SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM
POINT OF THE POTENTIAL
IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Abstract. We compute the scattering amplitude for Schrödinger operators at a critical
energy level, corresponding to the maximum point of the potential. We follow [30], using
Isozaki-Kitada’s representation formula for the scattering amplitude, together with results
from [5] in order to analyze the contribution of trapped trajectories.
Contents
1. Introduction 2
2. Assumptions and main results 4
3. Proof of the main resolvent estimate 10
4. Representation of the Scattering Amplitude 16
5. Computations before the critical point 19
5.1. Computation of u− in the incoming region 19
5.2. Computation of u− along γ
6. Computation of u− at the critical point 23
6.1. Study of the transport equations for the phases 25
6.2. Taylor expansions of ϕ+ and ϕ
6.3. Asymptotics near the critical point for the trajectories 34
6.4. Computation of the ϕkj ’s 36
7. Computations after the critical point 43
7.1. Stationary phase expansion in the outgoing region 43
7.2. Behaviour of the phase function Φ 47
Date: April 12, 2007.
2000 Mathematics Subject Classification. 81U20,35P25,35B38,35C20.
Key words and phrases. Scattering amplitude, critical energy, Schrödinger equation.
Acknowledgments: We would like to thank Johannes Sjöstrand for helpful discussions during the prepa-
ration of this paper. The first author also thanks Victor Ivrii for supporting visits to Université Paris Sud,
Orsay, and the Department of Mathematics at Orsay for the extended hospitality.
http://arxiv.org/abs/0704.1632v2
2 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
7.3. Integration with respect to time 48
Appendix A. Proof of Proposition 2.5 53
Appendix B. A lower bound for the resolvent 53
Appendix C. Lagrangian manifolds which are transverse to Λ± 55
Appendix D. Asymptotic behaviour of certain integrals 57
References 60
1. Introduction
We study the semiclassical behavior of scattering amplitude at energy E > 0 for Schrö-
dinger operators
(1.1) P (x, hD) = −h
∆ + V (x)
where V is a real valued C∞ function on Rn, which vanishes at infinity. We shall suppose
here that E is close to a critical energy level E0 for P , which corresponds to a non-degenerate
global maximum of the potential. Here, we address the case where this maximum is unique.
Let us recall that, if V (x) = O(〈x〉−ρ) for some ρ > (n + 1)/2, then for any ω 6= θ ∈ Sn−1
and E > 0, the problem
P (x, hD)u = Eu,
u(x, h) = ei
2Ex·ω/h +A(ω, θ,E, h)e
2E|x|/h
|x|(n−1)/2
+ o(|x|(1−n)/2) as x→ +∞, x|x| = θ,
has a unique solution. The scattering amplitude at energy E for the incoming direction ω
and the outgoing direction θ is the real number A(ω, θ,E, h).
For potentials that are not decaying that fast at infinity, it is not that easy to write down a
stationary formula for the scattering amplitude: If V (x) = O(〈x〉−ρ) for some ρ > 1, one can
define the scattering matrix at energy E using wave operators (see Section 4 below). Then,
writing
(1.2) S(E, h) = Id− 2iπT (E, h),
one can see that T (E, h) is a compact operator on L2(Sn−1), which kernel T (ω, θ,E, h) is
smooth out of the diagonal in Sn−1×Sn−1. Then, the scattering amplitude is given for θ 6= ω,
(1.3) A(ω, θ,E, h) = c(E))h(n−1)/2T (ω, θ,E, h),
where
(1.4) c(E) = −2π(2E)−
4 (2π)
2 e−i
(n−3)π
We proceed here as in [30], where D. Robert and H. Tamura have studied the semiclassical
behavior of the scattering amplitude for short range potentials at a non-trapping energy E .
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 3
An energy E is said to be non-trapping when K(E), the trapped set K(E) at energy E, is
empty. This trapped set is defined as
(1.5) K(E) =
(x, ξ) ∈ p−1(E), exp(tHp)(x, ξ) 6→ ∞ as t→ ±∞
where Hp is the Hamiltonian vector field associated to the principal symbol p(x, ξ) =
V (x) of the operator P . Notice that the scattering amplitude has been first studied, in the
semiclassical regime, by B. Vainberg [32] and Y. Protas [27] in the case of compactly supported
potential, and for non-trapping energies, where they obtained the same type of result.
Under the non-trapping assumption, and some other non-degeneracy condition (in fact our
assumption (A4) below), D. Robert and H. Tamura have shown that the scattering amplitude
has an asymptotic expansion with respect to h. The non-degeneracy assumption implies in
particular that there is a finite number N∞ of classical trajectories for the Hamiltonian p,
with asymptotic direction ω for t→ −∞ and asymptotic direction θ as t→ +∞. Robert and
Tamura’s result is the following asymptotic expansion for the scattering amplitude:
(1.6) A(ω, θ,E, h) =
iS∞j /h
aj,m(ω, θ,E)h
m +O(h∞), h→ 0,
where S∞j is the classical action along the corresponding trajectory. Also, they have computed
the first term in this expansion, showing that it can be given in terms of quantities attached
to the corresponding classical trajectory only.
There are also some few works concerning the scattering amplitude when the non-trapping
assumption is not fulfilled. In his paper [24], L. Michel has shown that, if there is no trapped
trajectory with incoming direction ω and outgoing direction θ (see the discussion after (2.6)
below), and if there is a complex neighborhood of E of size ∼ hN for some N ∈ N possibly
large, which is free of resonances, then A(ω, θ,E, h) is still given by Robert and Tamura’s
formula. The potential is also supposed to be analytic in a sector out of a compact set, and
the assumption on the existence of a resonance free domain around E amounts to an estimate
on boundary value of the meromorphic extension of the truncated resolvent of the for
(1.7) ‖χ(P − (E ± i0))−1χ‖ = O(h−N ), χ ∈ C∞0 (Rn).
Of course, these assumptions allow the existence of a non-empty trapped set.
In [2] and [3], the first author has shown that at non-trapping energies or in L. Michel’s
setting, the scattering amplitude is an h-Fourier Integral Operator associated to a natural
scattering relation. These results imply that the scattering amplitude admits an asymptotic
expansion even without the non-degeneracy assumption, and in the sense of oscillatory inte-
grals. In particular, the expansion (1.6) is recovered under the non-degeneracy assumption
and as an oscillatory integral.
In [21], A. Lahmar-Benbernou and A. Martinez have computed the scattering amplitude
at energy E ∼ E0, in the case where the trapped set K(E0) consists in one single point
corresponding to a local minimum of the potential (a well in the island situation). In that
case, the estimate (1.7) is not true, and their result is obtained through a construction of the
resonant states.
In the present work, we compute the scattering amplitude at energy E ∼ E0 in the case
where the trapped set K(E0) corresponds to the unique global maximum of the potential.
4 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
The one-dimensional case has been studied in [28, 14, 15], with specific techniques, and we
consider here the general n > 1 dimensional case.
Notice that J. Sjöstrand in [31], and P. Briet, J.-M. Combes and P. Duclos in [7, 8] have
described the resonances close to E0 in the case where V is analytic in a sector around R
From their result, it follows that Michel’s assumption on the existence of a not too small
resonance-free neighborhood of E0 is satisfied. However, we show below (see Proposition 2.5)
that for any ω ∈ Sn−1, there is at least one half-trapped trajectory with incoming direction
ω, so that L. Michel’s result never applies here.
Here, we do not assume analyticity for V . We compute the contributions to the scattering
amplitude arising from the classical trajectories reaching the unstable equilibrium point, which
corresponds to the top of the potential barrier. At the quantum level, tunnel effect occurs,
which permits the particle to pass through this point. Our computation here relies heavily
on [5], where a precise description of this phenomena has been obtained. In a forthcoming
paper, we shall show that in this case also, the scattering amplitude is an h-Fourier Integral
Operator.
This paper is organized in the following way. In Section 2, we describe our assumptions, and
state our main results: a resolvent estimate, and the asymptotic expansion of the scattering
amplitude in the semiclassical regime. Section 3 is devoted to the proof of the resolvent
estimate, from which we deduce in Section 4 estimates similar to those in [30]. In that
section, we also recall briefly the representation formula for the scattering amplitude proved
by Isozaki and Kitada, and introduce notations from [30]. The computation of the asymptotic
expansion of the scattering amplitude is conducted in sections 5, 6 and 7, following the classical
trajectories. Eventually, we have put in four appendices the proofs of some side results or
technicalities.
2. Assumptions and main results
We suppose that the potential V satisfies the following assumptions
(A1) V is a C∞ function on Rn, and, for some ρ > 1,
∂αV (x) = O(〈x〉−ρ−|α|).
(A2) V has a non-degenerate maximum point at x = 0, with E0 = V (0) > 0 and
∇2V (0) =
. . .
 , 0 < λ1 ≤ λ2 ≤ . . . ≤ λn.
(A3) The trapped set at energy E0 is K(E0) = {(0, 0)}.
Notice that the assumptions (A1)–(A3) imply that V has an absolute global maximum
at x = 0. Indeed, if L = {x 6= 0; V (x) ≥ E0} was non empty, the geodesic, for the
Agmon distance (E0 − V (x))1/2+ dx, between 0 and L would be the projection of a trapped
bicharacteristic (see [1, Theorem 3.7.7]).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 5
As in D. Robert and H. Tamura in [30], one of the key ingredient for the study of the
scattering amplitude is a suitable estimate for the resolvent. Using the ideas in [5, Section 4],
we have obtained the following result, that we think to be of independent interest.
Theorem 2.1. Suppose assumptions (A1), (A2) and (A3) hold, and let α > 1
be a fixed
real number. We have
(2.1) ‖P − (E ± i0))−1‖α,−α . h−1| lnh|,
uniformly for |E − E0| ≤ δ, with δ > 0 small enough. Here ‖Q‖α,β denotes the norm of the
bounded operator Q from L2(〈x〉α dx) to L2(〈x〉β dx).
Moreover, we prove in the Appendix B that our estimate is not far from optimal. Indeed,
we have the
Proposition 2.2. Under the assumptions (A1) and (A2), we have
(2.2) ‖(P − E0 ± i0)−1‖α,−α & h−1
| lnh|.
We would like to mention that in the case of a closed hyperbolic orbit, the same upper
bound has been obtained by N. Burq [9] in the analytic category, and in a recent paper [11]
by H. Christianson in the C∞ setting.
As a matter of fact, in the present setting, S. Nakamura has proved in [26] an O(h−2)
bound for the resolvent. Nakamura’s estimate would be sufficient for our proof of Theorem
2.6, but it is not sharp enough for the computation of the total scattering cross section along
the lines of D. Robert and H. Tamura in [29]. In that paper, the proof relies on a bound
O(h−1) for the resolvent, but it is easy to see that an estimate like O(h−1−ε) for any small
enough ε > 0 is sufficient. If we denote
(2.3) σ(ω,E0, h) =
|A(ω, θ,E, h)|2dθ,
the total scattering cross-section, and following D. Robert and H. Tamura’s work, our resolvent
estimates gives the
Theorem 2.3. Suppose assumptions (A1), (A2) and (A3) hold, and that ρ > n+1
, n ≥ 2.
If |E − E0| < δ for some δ > 0 small enough, then
(2.4) σ(ω,E, h) = 4
2−1(2E)−1/2h−1
V (y + sω)ds
dy +O(h−(n−1)/(ρ−1)).
Now we state our assumptions concerning the classical trajectories associated with the
Hamiltonian p, that is curves t 7→ γ(t, x, ξ) = exp(tHp)(x, ξ) for some initial data (x, ξ) ∈
T ∗Rn. Let us recall that, thanks to the decay of V at infinity, for given α ∈ Sn−1 and
z ∈ α⊥ ∼ Rn−1 (the impact plane), there is a unique bicharacteristic curve
(2.5) γ±(t, z, α,E) = (x±(t, z, α,E), ξ±(t, z, α,E))
such that
(2.6)
|x±(t, z, α,E) −
2Eαt− z| = 0,
|ξ±(t, z, α,E) −
2Eα| = 0.
6 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
We shall denote by Λ−ω the set of points in T
n lying on trajectories going to infinity with
direction ω as t → −∞, and Λ+θ the set of those which lie on trajectories going to infinity
with direction θ as t→ +∞:
(2.7)
Λ−ω =
γ−(t, z, ω,E) ∈ T ∗Rn, z ∈ ω⊥, t ∈ R
Λ+θ =
γ+(t, z, θ, E) ∈ T ∗Rn, z ∈ θ⊥, t ∈ R
We shall see that Λ−ω and Λ
θ are in fact Lagrangian submanifolds of T
Under the assumptions (A1), (A2) and (A3) there are only two possible behaviors for
x±(t, z, α,E) as t→ ∓∞: either it escapes to ∞, or it goes to 0.
First we state our assumptions for the first kind of trajectories. For these, we also have,
for some ξ∞(z, ω,E),
ξ−(t, z, ω) = ξ∞(z, ω,E),
and we shall say that the trajectory γ−(t, z, ω,E) has initial direction ω and final direction
θ = ξ∞(z, ω,E)/2
E. As in [30] we shall suppose that there is only a finite number of
trajectories with initial direction ω and final direction θ. This assumption can be given in
terms of the angular density
(2.8) σ̂(z) = |det(ξ∞(z, ω,E), ∂z1ξ∞(z, ω,E), . . . , ∂zn−1ξ∞(z, ω,E))|.
Definition 2.4. The outgoing direction θ ∈ Sn−1 is called regular for the incoming direction
ω ∈ Sn−1, or ω-regular, if θ 6= ω and, for all z′ ∈ ω⊥ with ξ∞(z′, ω,E) = 2
Eθ, the map
ω⊥ ∋ z 7→ ξ∞(z, ω,E) ∈ Sn−1 is non-degenerate at z′, i.e. σ̂(z′) 6= 0.
We fix the incoming direction ω ∈ Sn−1, and we assume that
(A4) the direction θ ∈ Sn−1 is ω-regular.
Then, one can show that Λ−ω ∩ Λ+θ is a finite set of Hamiltonian trajectories (γ∞j )1≤j≤N∞ ,
γ∞j (t) = γ
∞(t, z∞j ) = (x
j (t), ξ
j (t)), with transverse intersection along each of these curves.
We turn to trapped trajectories. Let us notice that the linearization Fp at (0, 0) of the
Hamilton vector field Hp has eigenvalues −λn, . . . ,−λ1, λ1, . . . , λn. Thus (0, 0) is a hyper-
bolic fixed point for Hp, and the stable/unstable manifold Theorem gives the existence of
a stable incoming Lagrangian manifold Λ− and a stable outgoing Lagrangian manifold Λ+
characterized by
(2.9) Λ± = {(x, ξ) ∈ T ∗Rn, exp(tHp)(x, ξ) → 0 as t→ ∓∞} .
In this paper, we shall describe the contribution to the scattering amplitude of the trapped
trajectories, that is those going from infinity to the fixed point (0, 0). We have proved in
Appendix A the following result, which shows that there are always such trajectories.
Proposition 2.5. For every ω, θ ∈ Sn−1, we have
(2.10) Λ−ω ∩ Λ− 6= ∅ and Λ+θ ∩ Λ+ 6= ∅.
We suppose that
(A5) Λ−ω and Λ− (resp. Λ
and Λ+) intersect transversally.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 7
Under this assumption, Λ−ω ∩ Λ− and Λ+θ ∩ Λ+ are finite sets of bicharacteristic curves. We
denote them, respectively,
(2.11) γ−k : t 7→ γ
−(t, z−k ) = (x
k (t), ξ
−(t)), 1 ≤ k ≤ N−,
(2.12) γ+ℓ : t 7→ γ
+(t, z+ℓ ) = (x
+(t), ξ+(t)), 1 ≤ ℓ ≤ N+.
Here, the z−
(resp. the z+
) belong to ω⊥ (resp. θ⊥) and determine the corresponding curve
by (2.6).
We recall from [18, Section 3] (see also [5, Section 5]), that each integral curve γ±(t) =
(x±(t), ξ±(t)) ∈ Λ± satisfies, in the sense of expandible functions (see Definition 6.1 below),
(2.13) γ±(t) ∼
γ±j (t)e
±µj t, as t→ ∓∞,
where µ1 = λ1 < µ2 < . . . is the strictly increasing sequence of linear combinations over N of
the λj’s. Here, the functions γ
j : R → R2n are polynomials, that we write
(2.14) γ±j (t) =
M ′j∑
γ±j,mt
Considering the base space projection of these trajectories, we denote
(2.15) x±(t) ∼
g±j (t)e
±µj t, as t→ ∓∞, g±j (t) =
M ′j∑
g±j,mt
Let us denote ̂ the (only) integer such that µb = 2λ1. We prove in Proposition 6.11 below that
if j < ̂, then M ′j = 0, or more precisely, that g
j (t) = g
j is a constant vector in Ker(Fp∓λj).
We also have M ′
≤ 1, and g−
can be computed in terms of g−1 .
In this paper, concerning the incoming trajectories, we shall assume that,
(A6) For each k ∈ {1, . . . , N−}, g−1 (z
) 6= 0.
Finally, we state our assumptions for the outgoing trajectories γ+ℓ ⊂ Λ+ ∩Λθ+. First of all,
it is easy to see, using Hartman’s linearization theorem, that there exists always a m ∈ N
such that g+m(z
ℓ ) 6= 0. We denote
(2.16) ℓℓℓ = ℓℓℓ(ℓ) = min{m, g+m(z+ℓ ) 6= 0}
the smallest of these m’s. We know that µℓℓℓ is one of the λj’s, and that M
ℓℓℓ = 0.
In [5], we have been able to describe the branching process between an incoming curve
γ− ⊂ Λ− and an outgoing curve γ+ ⊂ Λ+ provided 〈g−1 |g
1 〉 6= 0 (see the definition for
Λ̃+(ρ−) before [5, Theorem 2.6]). Here, for the computation of the scattering amplitude, we
can relax a lot this assumption, and analyze the branching in other cases that we describe
now. Let us denote, for a given pair of paths (γ−(z−k ), γ
+(z+ℓ )) in (Λ
ω ∩ Λ−)× (Λ+θ ∩ Λ+),
(2.17) M2(k, ℓ) = −
j∈I1(2λ1)
α,β∈I2(λ1)
βV (0)
(g−1 (z
αV (0)
(g+1 (z
8 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
M1(k, ℓ) =−
α∈I2(λ1)
αV (0)
(g−1 (z
−))α(g+
(z+))j + (g
(z−))j(g
α,β∈I2(λ1)
(g−1 (z
(g+1 (z
Cα,β,(2.18)
where
Cα,β =− ∂α+βV (0) +
j∈I1\I1(2λ1)
λ2j(4λ
1 − λ2j)
∂α+γV (0)∂β+γV (0)
γ,δ∈I2(λ1)
γ+δ=α+β
(γ + δ)!
γ! δ!
γV (0)∂j∂
δV (0).(2.19)
Here, we have set I1 = {1, . . . , n}, 1j = (δij)i=1,...,n ∈ Nn and
(2.20) Im(µ) = {β ∈ Nn, β = 1k1 + · · ·+ 1km with λk1 = · · · = λkm = µ},
the set of multi-indices β of length |β| = m with each index of its non-vanishing components
in the set {j ∈ N, λj = µ}. We also denote Im ⊂ Nn the set of multi-indices of length m.
We will suppose that
(A7) For each pair of paths (γ−(z−k ), γ
+(z+ℓ )), k ∈ {1, . . . , N−}, ℓ ∈ {1, . . . , N+}, one of the
three following cases occurs:
(a) The set
m < ̂, 〈g−m(z−k )|g+m(z
ℓ )〉 6= 0
is not empty. Then we denote
k = min
m < ̂, 〈g−m(z−k )|g
)〉 6= 0
(b) For all m < ̂, we have 〈g−m(z−k )|g+m(z
ℓ )〉 = 0, and M2(k, ℓ) 6= 0.
(c) For all m < ̂, we have 〈g−m(z−k )|g+m(z
ℓ )〉 = 0, M2(k, ℓ) = 0 and M1(k, ℓ) 6= 0.
As one could expect (see [30], [28] or [15]), action integrals appear in our formula for the
scattering amplitude. We shall denote
S∞j =
(|ξ∞j (t)|2 − 2E0)dt, j ∈ {1, . . . , N∞},(2.21)
S−k =
|ξ−k (t)|
2 − 2E01t<0
dt, k ∈ {1, . . . , N−},(2.22)
(t)|2 − 2E01t>0
dt, ℓ ∈ {1, . . . , N+},(2.23)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 9
and ν∞j , ν
ℓ , ν
k the Maslov indexes of the curves γ
j , γ
ℓ , γ
k respectively. Let also
D−k = limt→+∞
∣∣∣ det
∂x−(t, z, ω,E0)
∂(t, z)
∣∣∣ e−(Σλj−2λ1)t,(2.24)
D+ℓ = limt→−∞
∣∣∣ det
∂x+(t, z, ω,E0)
∂(t, z)
∣∣∣ e(Σλj−2λℓℓℓ)t,(2.25)
be the Maslov determinants for γ−
, and γ+
respectively. We show below that 0 < D−
+∞. Eventually we set
(2.26) Σ(E, h) =
− iE − E0
Then, the main result of this paper is the
Theorem 2.6. Suppose assumptions (A1) to (A7) hold, and that E ∈ R is such that
E −E0 = O(h). Then
A(ω, θ,E, h) =
Aregj (ω, θ,E, h) +
Asingk,ℓ (ω, θ,E, h) +O(h
∞),(2.27)
where
(2.28) Aregj (ω, θ,E, h) = e
iS∞j /h
j,m(ω, θ,E)h
j,0 (ω, θ,E) =
−iν∞j π/2
σ̂(zj)1/2
Moreover we have
• In case (a)
Asingk,ℓ (ω, θ,E, h) = e
k,ℓ,m(ω, θ,E, lnh)h
(Σ(E)+bµm)/µk−1/2,(2.29)
where the a
k,ℓ,m
(ω, θ,E, ln h) are polynomials with respect to lnh, and
k,ℓ,0(ω, θ,E, ln h) =
π1−n/2
ei(nπ/4−π/2)
)−1/2
(Σ(E)
(2λ1λℓℓℓ)
× e−iν
π/2e−iν
π/2(D−
)−1/2
× |g−1 (z
k )| |g
ℓℓℓ (z
(z−k )
(z+ℓ )
〉)−Σ(E)/µk .(2.30)
• In case (b)
(2.31) Asingk,ℓ (ω, θ,E, h) = e
k,ℓ (ω, θ,E)
hΣ(E)/2λ1−1/2
| lnh|Σ(E)/λ1
(1 + o(1)),
10 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where
(ω, θ,E) =
π1−n/2
ei(nπ/4−π/2)
)−1/2
(Σ(E)
(2λ1λℓℓℓ)
3/2(2λ1)
Σ(E)/λ1−1
× e−iν
π/2e−iν
π/2(D−k D
× |g−1 (z
)| |g+
− iM2(k, ℓ)
)−Σ(E)/2λ1
.(2.32)
• In case (c)
(2.33) Asingk,ℓ (ω, θ,E, h) = e
k,ℓ (ω, θ,E)
hΣ(E)/2λ1−1/2
| ln h|Σ(E)/2λ1
(1 + o(1)),
where
k,ℓ (ω, θ,E) =
π1−n/2
ei(nπ/4−π/2)
)−1/2
(Σ(E)
(2λ1λℓℓℓ)
3/2(2λ1)
Σ(E)/2λ1−1
× e−iν
π/2e−iν
π/2(D−k D
× |g−1 (z
k )| |g
ℓℓℓ (z
− iM1(k, ℓ)
)−Σ(E)/2λ1
.(2.34)
Here, the µ̂j are the linear combinations over N of the λk’s and λk − λ1’s, and the function
z 7→ z−Σ(E)/µk is defined on C\]−∞, 0] and real positive on ]0,+∞[.
Of course the assumption that 〈g−1 |g
1 〉 6= 0 (a subcase of (a)) is generic. Without the
assumption (A4), the regular part Areg of the scattering amplitude has an integral rep-
resentation as in [3]. When the assumption (A7) is not fulfilled, that is when the terms
corresponding to the µj with j ≤ ̂ do not contribute, we don’t know if the scattering
amplitude can be given only in terms of the g±1 ’s and of the derivatives of the potential.
3. Proof of the main resolvent estimate
Here we prove Theorem 2.1 using Mourre’s Theory. We start with the construction of an
escape function close to the stationary point (0, 0) in the spirit of [10] and [5]. Since Λ+ and
Λ− are Lagrangian manifolds, one can choose local symplectic coordinates (y, η) such that
(3.1) p(x, ξ) = B(y, η)y · η,
where (y, η) 7→ B(y, η) is a C∞ mapping from a neighborhood of (0,0) in T ∗Rn to the space
Mn(R) of n× n matrices with real entries, such that,
(3.2) B(0, 0) =
. . .
We denote U a unitary Fourier Integral Operator (FIO) microlocally defined in a neighborhood
of (0, 0), which canonical transformation is the map (x, ξ) 7→ (y, η), and we set
(3.3) P̂ = UPU∗.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 11
Here the FIO U∗ is the adjoint of U , and we have UU∗ = Id+O(h∞) and U∗U = Id+O(h∞)
microlocally near (0, 0). Then P̂ is a pseudodifferential operator, with a real (modulo O(h∞))
symbol p̂(y, η) =
j p̂j(y, η)h
j , such that
(3.4) p̂0 = B(y, η)y · η.
We set B1 = Oph(b1),
(3.5) b1(y, η) =
χ̃2(y, η),
where M > 1 will be fixed later and χ̃1 ≺ χ̃2 ∈ C∞0 (T ∗Rn) with χ̃1 = 1 near (0, 0). In what
follows, we will assume that hM < 1. In particular, b1 ∈ S1/2(| ln h|). Here and in what
follows, we use the usual notation for classes of symbols. For m an order function, a function
a(x, ξ, h) ∈ C∞(T ∗Rn) belongs to Sδh(m) when
(3.6) ∀α ∈ N2n, ∃Cα > 0, ∀h ∈]0, 1], |∂αx,ξa(x, ξ, h)| ≤ Cαh−δ|α|m(x, ξ).
Let us also recall that, if a ∈ Sα(1) and b ∈ Sβ(1), with α, β < 1/2, we have
(3.7)
Oph(a),Oph(b)
= Oph
ih{b, a}
+ h3(1−α−β) Oph(r),
with r ∈ Smin(α,β)(1): In particular the term of order 2 vanishes.
Hence, we have here
(3.8) [B1, P̂ ] = Oph
ih{p̂0, b1}
+ | lnh|h3/2 Oph(rM ),
with rM ∈ S1/2(1). The semi-norms of rM may depend on M . We have
(3.9) {p̂0, b1} = c1 + c2,
{p̂0, χ̃2}(3.10)
p̂0, ln
By + (∂ηB)y · η
hM + y2
Bη + (∂yB)y · η
hM + η2
χ̃2.(3.11)
The symbols c1 ∈ S1/2(| lnh|), c2 ∈ S1/2(1) satisfy supp(c1) ⊂ supp(∇χ̃2). Let ϕ̃ ∈
C∞0 (T
n) be a function such that ϕ̃ = 0 near (0, 0) and ϕ̃ = 1 near the support of ∇χ̃2. We
Oph(c1) =Oph(ϕ̃)Oph(c1)Oph(ϕ̃) +O(h∞)
≥− C1h| ln h|Oph(ϕ̃)Oph(ϕ̃) +O(h∞)
≥− C1h| ln h|Oph(ϕ̃2) +O(h2| ln h|),(3.12)
for some C1 > 0. On the other hand, using [5, (4.96)–(4.97)], we get
(3.13) Oph(c2) ≥ εM−1 Oph(χ̃1) +O(M−2),
12 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
for some ε > 0. With the notation A1 = U
∗B1U , the formulas (3.8), (3.9), (3.12) and (3.13)
imply
−i[A1, P ] =− iU∗[B1, P ]U
≥εhM−1U∗Oph(χ̃1)U − C1h| lnh|U∗ Oph(ϕ̃2)U
+O(hM−2) +OM (h3/2| lnh|).(3.14)
If κ is the canonical transformation associated to U , then χj = χ̃j ◦ κ, j = 1, 2 and ϕ = ϕ̃ ◦ κ
are C∞0 (T
∗(Rn), [0, 1]) functions which satisfy χ1 = 1 near (0, 0) and ϕ = 0 near (0, 0). Using
Egorov’s Theorem, (3.14) becomes
(3.15) − i[A1, P ] ≥ εhM−1 Oph(χ1)− C1h| lnh|Oph(ϕ) +O(hM−2) +OM (h3/2| lnh|).
Now, we build an escape function outside of supp(χ1) as in [22]. Let 1(0,0) ≺ χ0 ≺
χ1 ≺ χ2 ≺ χ3 ≺ χ4 ≺ χ5 be C∞0 (T ∗(Rn), [0, 1]) functions with ϕ ≺ χ4. We define a3 =
g(ξ)(1−χ3(x, ξ))x ·ξ where g ∈ C∞0 (Rn) satisfies 1p−1([E0−δ,E0+δ]) ≺ g. Using [6, Lemma 3.1],
we can find a bounded, C∞ function a2(x, ξ) such that
(3.16) Hpa2 ≥
0 for all (x, ξ) ∈ p−1([E0 − δ,E0 + δ]),
1 for all (x, ξ) ∈ supp(χ4 − χ0) ∩ p−1([E0 − δ,E0 + δ]),
and we set A2 = Oph(a2χ5). We denote
(3.17) A = A1 + C2| lnh|A2 + | lnh|A3,
where C2 > 1 will be fixed later. Now let ψ̃ ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]) with ψ̃ = 1 near
E0. We recall that ψ̃(P ) is a classical pseudodifferential operator of class Ψ
0(〈ξ〉−∞) with
principal symbol ψ̃(p). Then, from (3.15), we obtain
−iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1ψ̃(P )Oph(χ1)ψ̃(P )− C1h| ln h|ψ̃(P )Oph(ϕ)ψ̃(P )
+ C2h| lnh|Oph
ψ̃2(p)(χ4 − χ0)
+ C2h| ln h|Oph
ψ̃2(p)a2Hpχ5
+ h| ln h|Oph
ψ̃2(p)(ξ2 − x · ∇V )(1 − χ3)
+ h| ln h|Oph
ψ̃2(p)x · ξHp(gχ3)
+O(hM−2) +OM (h3/2| lnh|).(3.18)
From (A1), we have x·∇V (x) → 0 as x→ ∞. In particular, if χ3 is equal to 1 in a sufficiently
large zone, we have
(3.19) ψ̃2(p)(ξ2 − x · ∇V )(1− χ3) ≥ E0ψ̃2(p)(1− χ3).
If C2 > 0 is large enough, the G̊arding inequality implies
(3.20)
C2 Oph
ψ̃2(p)(χ4 − χ0)
−C1 Oph
ψ̃2(p)ϕ
ψ̃2(p)x · ξHp(gχ3)
≥ Oph
ψ̃2(p)(χ4 − χ0)
+O(h).
As in [22], we take χ5(x) = χ̃5(µx) with µ small and χ̃5 ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]). Since
a2 is bounded, we get
(3.21)
∣∣C2ψ̃2(p)a2Hpχ5
∣∣ ≤ µC2‖a2‖L∞‖Hpχ̃5‖L∞ . µ.
Therefore, if µ is small enough, (3.19) implies
(3.22) Oph
ψ̃2(p)(ξ2−x ·∇V )(1−χ3)
+C2 Oph
ψ̃2(p)a2Hpχ5
ψ̃2(p)(1−χ3)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 13
Then (3.18), (3.20), (3.22) and the G̊arding inequality give
−iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1 Oph
ψ̃2(p)χ1
+ h| ln h|Oph
ψ̃2(p)(χ4 − χ0)
h| ln h|Oph
ψ̃2(p)(1 − χ3)
+O(hM−2) +OM (h3/2| lnh|)
≥εhM−1 Oph
ψ̃2(p)
+O(hM−2) +OM (h3/2| ln h|).(3.23)
Choosing M large enough and 1E0 ≺ ψ ≺ ψ̃, we have proved the
Lemma 3.1. Let M be large enough and ψ ∈ C∞0 ([E0− δ,E0+ δ]), δ > 0 small enough, with
ψ = 1 near E0. Then, we have
(3.24) − iψ(P )[A,P ]ψ(P ) ≥ εh−1ψ2(P ).
Moreover
(3.25) [A,P ] = O(h| ln h|).
From the properties of the support of the χj, we have
[[P,A], A] =[[P,A1], A1] + C2| lnh|[[P,A1], A2]
+ C2| ln h|[[P,A2], A1] + C22 | lnh|2[[P,A2], A2] + C2| lnh|2[[P,A2], A3]
+ C2| ln h|2[[P,A3], A2] + | lnh|2[[P,A3], A3] +O(h∞).(3.26)
We also know that P ∈ Ψ0(〈ξ〉2), A2 ∈ Ψ0(〈ξ〉−∞) and A3 ∈ Ψ0(〈x〉〈ξ〉−∞). Then, we can
show that all the terms in (3.26) with j, k = 2, 3 satisfy
(3.27) [[P,Aj ], Ak] ∈ Ψ0(h2).
On the other hand,
(3.28) [[P,A1], A2] = U
∗[[P̂ , B1], UA2U
∗]U +O(h∞),
with UA2U
∗ ∈ Ψ0(1). From (3.8) – (3.11), we have [P̂ , B1] ∈ Ψ1/2(h| ln h|) and then
(3.29) [[P,A1], A2] = O(h3/2| lnh|).
The term [[P,A2], A1] gives the same type of contribution. It remains to study
(3.30) [[P,A1], A1] = U
∗[[P̂ , B1], B1]U +O(h∞).
Let χ̃3 ∈ C∞0 (T ∗Rn), [0, 1]) with χ̃2 ≺ χ̃3 and
(3.31) f =
χ̃3(y, η) ∈ S1/2(| ln h|).
Then, with a remainder rM ∈ S1/2(1) which differs from line to line,
i[P̂ , B1] =hOph
f{χ̃2, p̂0}+ c2
− h3/2| ln h|Oph(rM )
=hOph(f)Oph({χ̃2, p̂0}) + hOph(c2) + h3/2| lnh|Oph(rM ).(3.32)
14 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
In particular, since [P̂ , B1] ∈ Ψ1/2(h| ln h|), c2 ∈ S1/2(1) and f ∈ S1/2(| lnh|),
[[P̂ , B1], B1] =[[P̂ , B1],Oph(fχ̃2)]
=− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(fχ̃2)]− ih[Oph(c2),Oph(fχ̃2)]
+O(h3/2| ln h|2)
=− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(f)Oph(χ̃2)] +O(h| ln h|)
=− ihOph(f)[Oph({χ̃2, p̂0}),Oph(f)]Oph(χ̃2)
− ih[Oph(f),Oph(f)]Oph({χ̃2, p̂0})Oph(χ̃2)
− ihOph(f)Oph(f)[Oph({χ̃2, p̂0}),Oph(χ̃2)]
− ihOph(f)[Oph(f),Oph(χ̃2)]Oph({χ̃2, p̂0}) +O(h| ln h|)
=O(h| ln h|).(3.33)
From (3.26), (3.27), (3.29) and (3.33), we get
(3.34) [[P,A], A] = O(h| lnh|).
As a matter of fact, using [5], one can show that [[P,A], A] = O(h). Now we can use the
following proposition which is an adaptation of the limiting absorption principle of Mourre
[25] (see also [12, Theorem 4.9], [19, Proposition 2.1] and [4, Theorem 7.4.1]).
Proposition 3.2. Let (P,D(P )) and (A,D(A)) be self-adjoint operators on a separable
Hilbert space H. Assume the following assumptions:
i) P is of class C2(A). Recall that P is of class Cr(A) if there exists z ∈ C \ σ(P ) such
(3.35) R ∋ t→ eitA(P − z)−1e−itA,
is Cr for the strong topology of L(H).
ii) The form [P,A] defined on D(A) ∩D(P ) extends to a bounded operator on H and
(3.36) ‖[P,A]‖ . β.
iii) The form [[P,A],A] defined on D(A) extends to a bounded operator on H and
(3.37) ‖[[P,A],A]‖ . γ.
iv) There exist a compact interval I ⊂ R and g ∈ C∞0 (R) with 1I ≺ g such that
(3.38) ig(P )[P,A]g(P ) & γg2(P ).
v) β2 . γ . 1.
Then, for all α > 1/2, limε→0〈A〉−α(P − E ± iε)−1〈A〉−α exists and
(3.39)
∥∥〈A〉−α(P − E ± i0)−1〈A〉−α
∥∥ . γ−1,
uniformly for E ∈ I.
Remark 3.3. From Theorem 6.2.10 of [4], we have the following useful characterization of
the regularity C2(A). Assume that (ii) and (iv) hold. Then, P is of class C2(A) if and only
if, for some z ∈ C \ σ(P ), the set {u ∈ D(A); (P − z)−1u ∈ D(A) and (P − z)−1u ∈ D(A)}
is a core for A.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 15
Proof. The proof follows the work of Hislop and Nakamura [19]. For ε > 0, we define M2 =
ig(P )[P,A]g(P ) and Gε(z) = (P − iεM2 − z)−1 which is analytic for Re z ∈ I and Im z > 0.
Following [12, Lemma 4.14] with (3.35)), we get
(3.40) ‖g(P )Gε(z)ϕ‖ . (εγ)−1/2|(ϕ,Gε(z)ϕ)|1/2,
(3.41) ‖(1− g(P ))Gε(z)‖ . 1 + εβ‖Gε(z)‖,
and then
(3.42) ‖Gε(z)‖ . (εγ)−1,
for ε < ε0 with ε0 small enough, but independent on β, γ.
As in [19], let Dε = (1 + |A|)−α(1 + ε|A|)α−1 for α ∈]1/2, 1] and Fε(z) = DεGε(z)Dε. Of
course, from (3.42),
(3.43) ‖Fε(z)‖ . (εγ)−1,
and (3.40) and (3.41) with ϕ = Dεψ give
(3.44) ‖Gε(z)Dε‖ . 1 + (εγ)−1/2‖Fε‖1/2.
The derivative of Fε(z) is given by (see [12, Lemma 4.15])
(3.45) ∂εFε(z) = iDεGεM
2GεDε = Q0 +Q1 +Q2 +Q3,
Q0 =(α− 1)|A|(1 + |A|)−α(1 + ε|A|)α−2Gε(z)Dε
+ (α− 1)DεGε(z)|A|(1 + |A|)−α(1 + ε|A|)α−2(3.46)
Q1 =DεGε(1− g(P ))[P,A](1 − g(P ))GεDε(3.47)
Q2 =DεGε(1− g(P ))[P,A]g(P )GεDε +DεGεg(P )[P,A](1 − g(P ))GεDε(3.48)
Q3 =−DεGε[P,A]GεDε.(3.49)
From (3.44), we obtain
(3.50) ‖Q0‖ . εα−1
1 + (εγ)−1/2‖Fε‖1/2
and from (3.36), v) of Proposition 3.2, (3.41) and (3.42), we get
(3.51) ‖Q1‖ . γ−1.
Using in addition (3.44), we obtain
(3.52) ‖Q2‖ . 1 + (εγ)−1/2‖Fε‖1/2.
Now we write Q3 = Q4 +Q5 with
Q4 = −DεGε[P − iεM2 − z,A]GεDε(3.53)
Q5 = −iεDεGε[M2,A]GεDε.(3.54)
For Q4, we have the estimate
(3.55) ‖Q4‖ . εα−1
1 + (εγ)−1/2‖Fε‖1/2
On the other hand, (3.36), (3.37) and v) imply
(3.56) ‖[M2,A]‖ . γ.
16 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Then (3.44) gives
(3.57) ‖Q5‖ . 1 + ‖Fε‖.
Using the estimates on the Qj, we get
(3.58) ‖∂εFε‖ . εα−1
γ−1 + (εγ)−1/2‖Fε‖1/2 + ‖Fε‖
Using (3.43) and integrating (3.37) N times with respect to ε, we get
(3.59) ‖Fε‖ . γ−1
1 + ε2α(1−2
−N )−1),
so that, for N large enough,
(3.60) lim sup
‖〈A〉−α(P − E ± iδ)−1〈A〉−α‖ . γ−1.
Using, as in [19], that z 7→ F0(z) is Hölder continuous, we prove the existence of the limit
limIm z→0 F0(z) for Re z ∈ I and the proposition follows from (3.60). �
From Lemma 3.1 and (3.34), we can apply Proposition 3.2 with A = A/| ln h|, β = h and
γ = h/| ln h|. Therefore we have the estimate
(3.61)
∥∥〈A〉−α(P − E ± i0)−1〈A〉−α
∥∥ . h−1| ln h|,
for E ∈ [E0 − δ,E0 + δ]. As usual, we have
(3.62) ‖〈x〉−α〈A〉α‖ = O(1),
for α ≥ 0. Indeed, (3.62) is clear for α ∈ 2N, and the general case follows by complex
interpolation. Then, (3.61) and (3.26) imply Theorem 2.1.
4. Representation of the Scattering Amplitude
As in [30], our starting point for the computation of the scattering amplitude is the rep-
resentation given by Isozaki and Kitada in [20]. We recall briefly their formula, that they
obtained writing parametrices for the wave operators W± as Fourier Integral Operators, tak-
ing advantage of the well-known intertwining property W±P = P0W±, P = P0 + V . The
wave operators are defined by
(4.1) W± = s− lim
eitP/he−itP0/h,
where the limit exist thanks to the short-range assumption (A1). The scattering operator
is by definition S = (W+)∗W−, and the scattering matrix S(E, h) is then given by the
decompostion of S with respect to the spectral measure of P0 = −h2∆. Now we recall briefly
the discussion in [30, Section 1,2] (see also [3]), and we start with some notations.
If Ω is an open subset of T ∗Rn , we denote by Am(Ω) the class of symbols a such that
(x, ξ) 7→ a(x, ξ, h) belongs to C∞(Ω) and
(4.2)
∣∣∣∂αx ∂
ξ a(x, ξ)
∣∣∣ ≤ Cαβ〈x〉m−|α|〈ξ〉−L, for all L > 0, (x, ξ) ∈ Ω, (α, β) ∈ Nd × Nd.
We also denote by
(4.3) Γ±(R, d, σ) =
(x, ξ) ∈ Rn × Rn : |x| > R, 1
< |ξ| < d,± cos(x, ξ) > ±σ
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 17
with R > 1, d > 1, σ ∈ (−1, 1), and cos(x, ξ) = 〈x,ξ〉|x| |ξ| , the outgoing and incoming subsets
of T ∗Rn, respectively. Eventually, for α > 1
, we denote the bounded operator F0(E, h) :
L2α(R
n) → L2(Sn−1) given by
(4.4) (F0(E, h)f) (ω) = (2πh)−
2 (2E)
2E〈ω,x〉f(x)dx,E > 0.
Isozaki and Kitada have constructed phase functions Φ± and symbols a± and b± such that,
for some R0 >> 0, 1 < d4 < d3 < d2 < d1 < d0, and 0 < σ4 < σ3 < σ2 < σ1 < σ0 < 1:
i) Φ± ∈ C∞(T ∗Rn) solve the eikonal equation
(4.5)
|∇xΦ±(x, ξ)|2 + V (x) =
in (x, ξ) ∈ Γ±(R0, d0,±σ0), respectively.
ii) (x, ξ) 7→ Φ±(x, ξ) − x · ξ ∈ A0 (Γ±(R0, d0,±σ0)) .
iii) For all (x, ξ) ∈ T ∗Rn
(4.6)
∂xj∂ξk
(x, ξ) − δjk
∣∣∣∣ < ε(R0),
where δjk is the Kronecker delta and ε(R0) → 0 as R0 → +∞.
iv) a± ∼
j=0 h
ja±j , where a±j ∈ A−j(Γ±(3R0, d1,∓σ1)), supp a±j ⊂ Γ±(3R0, d1,∓σ1),
a±j solve
(4.7) 〈∇xΦ±|∇xa±0〉+
(∆xΦ±) a±0 = 0
(4.8) 〈∇xΦ±|∇xa±j〉+
(∆xΦ±) a±j =
∆xa±j−1, j ≥ 1,
with the conditions at infinity
(4.9) a±0 → 1, a±j → 0, j ≥ 1, as |x| → ∞.
in Γ±(2R0, d2,∓σ2), and solve (4.7) and (4.8) in Γ±(4R0, d1,∓σ2).
v) b± ∼
j=0 h
jb±j, where b±j ∈ A−j(Γ±(5R0, d3,±σ4), supp b±j ⊂ Γ±(5R0, d3,±σ4),
b±j solve (4.7) and (4.8) with the conditions at infinity (4.9) in Γ±(6R0, d4,±σ3), and
solve (4.7) and (4.8) in Γ±(6R0, d3,±σ3).
For a symbol c and a phase function ϕ, we denote by Ih(c, ϕ) the oscillatory integral
(4.10) Ih(c, ϕ) =
(2πh)n
(ϕ(x,ξ)−〈y,ξ〉)c(x, ξ)dξ
and we set
(4.11)
K±a(h) = P (h)Ih(a±,Φ±)− Ih(a±,Φ±)P0(h),
K±b(h) = P (h)Ih(b±,Φ±)− Ih(b±,Φ±)P0(h).
The operator T (E, h) for E ∈] 1
[ is then given by (see [20, Theorem 3.3])
(4.12) T (E, h) = T+1(E, h) + T−1(E, h) − T2(E, h),
18 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where
(4.13) T±1(E, h) = F0(E, h)Ih(a±,Φ±)∗K±b(h)F∗0 (E, h)
(4.14) T2(E, h) = F0(E, h)K∗+a(h)R(E + i0, h) (K+b(h) +K−b(h))F∗0 (E, h),
where we denote from now on R(E ± i0, h) = (P − (E ± i0))−1.
Writing explicitly their kernel, it is easy to see, by a non-stationary phase argument, that
the operators T±1 are O(h∞) when θ 6= ω. Therefore we have
(4.15) A(ω, θ,E, h) = −c(E)h(n−1)/2T2(ω, θ,E, h) +O(h∞),
where c(E) is given in (1.4).
As in [30], we shall use our resolvent estimate (Theorem 2.1) in a particular form. It was
noticed by L. Michel in [24, Proposition 3.1] that, in the present trapping case, the following
proposition follows easily from the corresponding one in the non-trapping setting. Indeed, if
ϕ is a compactly supported smooth function, it is clear that P̃ = −h2∆+ (1− ϕ(x/R))V (x)
satisfies the non-trapping assumption for R large enough, thanks to the decay of V at ∞.
Writing [30, Lemma 2.3] for P̃ , one gets the
Proposition 4.1. Let ω± ∈ A0 has support in Γ±(R, d, σ±) for R > R0. For E ∈ [E0 −
δ,E0 + δ], we have
(i) For any α > 1/2 and M > 1, then, for any ε > 0,
(4.16) ‖R(E ± i0, h)ω±(x, hDx)‖−α+M,−α = O(h−3−ε).
(ii) If σ+ > σ−, then for any α≫ 1,
(4.17) ‖ω∓(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞).
(iii) If ω(x, ξ) ∈ A0 has support in |x| < (9/10)R, then for any α≫ 1
(4.18) ‖ω(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞).
Then we can follow line by line the discussion after Lemma 2.1 of D. Robert and H. Tamura,
and we obtain (see Equations 2.2-2.4 there):
(4.19) A(ω, θ,E, h) = c(E)h−(n+1)/2〈R(E + i0, h)g−eiψ−/h, g+eiψ+/h〉+O(h∞),
where
(4.20) g± = e
−iψ±/h[χ±, P ]a±(x, h)e
iψ±/h,
(4.21) ψ+(x) = Φ+(x,
2Eθ), ψ−(x) = Φ−(x,
2Eω).
Moreover the functions χ± are C
n) functions such that χ± = 1 on some ball B(0, R±),
with support in B(0, R± + 1).
Eventually, we shall need the following version of Egorov’s Theorem, which is also used in
Robert and Tamura’s paper.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 19
Proposition 4.2 ([30, Proposition 3.1]). Let ω(x, ξ) ∈ A0 be of compact support. Assume
that, for some fixed t ∈ R, ωt is a function in A0 which vanishes in a small neighborhood of
{(x, ξ); (x, ξ) = exp(tHp)(y, η), (y, η) ∈ suppω}.
‖Oph(ωt)e−itP/hOph(ω)‖−α,α = O(h∞),
for any α ≫ 1. Moreover, the order relation is uniform in t when t ranges over a compact
interval of R.
In the three next sections, we prove Theorem 2.6 using (4.19). We set
(4.22) u− = u
− = R(E + i0, h)g−eiψ−/h,
and our proof consists in the computation of u− in different region of the phase space, following
the classical trajectories γ∞j , or γ
k and γ
ℓ . It is important to notice that we have (P−E)u− =
0 out of the support of g−.
5. Computations before the critical point
5.1. Computation of u− in the incoming region.
We start with the computation of u− in an incoming region which contains the micro-
support of g−. Notice that, thanks to Theorem 2.1, 〈x〉−αu−(x) is a semiclassical family of
distributions for α > 1/2.
Lemma 5.1. Let P be a Schrödinger operator as in (1.1) satisfying only (A1). Suppose that
I is a compact interval of ]0,+∞[, and d > 0 is such that I ⊂] 1
[. Suppose also that
0 < σ+ < 1, R is large enough and K ⊂ T ∗Rn is a compact subset of {|x| > R} ∩ p−1(I).
Then there exists T0 > 0 such that, if ρ ∈ K and t > T0,
(5.1) exp(tHp)(ρ) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn).
Proof. Let δ > 0. From the construction of C. Gérard and J. Sjöstrand [17], there exists a
function G(x, ξ) ∈ C∞(R2n) such that,
(HpG)(x, ξ) ≥ 0 for all (x, ξ) ∈ p−1(]
[),(5.2)
(HpG)(x, ξ) > 2E(1− δ) for |x| > R0 and p(x, ξ) = E ∈]
[,(5.3)
G(x, ξ) = x · ξ for |x| > R0.(5.4)
Let ρ ∈ K, and γ(t) = (x(t), ξ(t)) = exp(tHp)(ρ) be the corresponding Hamiltonian curve.
We distinguish between 2 cases:
1) For all t > 0, we have |x(t)| > R0.
Then G(γ(t)) > 2E(1 − δ)t+G(ρ) and, for t > T1 with T1 large enough,
(5.5) G(γ(t)) > 2 sup
x∈B(0,R0)
p(x,ξ)∈I
G(x, ξ).
20 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
By continuity, there exists a neighborhood U of γ such that, for all γ̃ ∈ U , we have
(5.6) G(γ̃(T1)) > sup
x∈B(0,R0)
p(x,ξ)∈I
G(x, ξ).
Since G is non-decreasing on γ̃(t), we have |x̃(t)| > R0 for all t > T1, and then
(5.7) G(γ̃(t)) > 2E(1 − δ)(t − T1) +G(γ̃(T1)) > 2E(1 − δ)t − C.
On the other hand, by uniformly finite propagation, we have |x̃(t)| <
2E(1+ δ)t+C. From
(5.7), we get |x̃(t)| > 1
t− C for all γ̃ ∈ U , and then |ξ̃(t)| =
2E + ot→∞(1). In particular,
the previous estimates gives
(5.8) |x(t)| > R/2,
(5.9) cos
x̃, ξ̃
(t) >
2E(1− δ)t− C
2E(1 + δ)t+ C)(
2E + ot→∞(1))
1 + δ
+ ot→∞(1) > 1− 3δ,
for t > T0 with T0 large enough but independent on γ̃ ∈ U . Thus, for t > T0 and γ̃ ∈ U , we
(5.10) γ̃(t) ∈ Γ+(R/2, d, σ+),
with σ+ = 1− 3δ.
2) There exist T2 > 0 such that |x(T2)| = R0.
Then there exists V a neighborhood of γ such that, for all γ̃ ∈ V, we have |x̃(T2)| < 2R0. Let
t > T2.
a) If |x̃(t)| ≤ R/2, then γ̃(t) ∈ B(0, R/2) × Rn.
b) Assume now |x̃(t)| > R/2. Denote by T3 (> T2) the last time (before t) such that
|x̃(T3)| = 2R0. Then
G(γ̃(t)) >2E(1 − δ)(t− T3) +G(γ̃(T3))(5.11)
>2E(1 − δ)(t− T3)− C,(5.12)
where C depend only on R0. On the other hand, the have |x̃(t)| <
2E(1 + δ)(t − T3) + C
(where the constant C depend only on R0). Then,
(5.13) t− T3 >
|x̃(t)|√
2E(1 + δ)
2E(1 + δ)
(5.14) |ξ̃(t)| =
2E + oR→∞(1),
x̃, ξ̃
(t) >
2E(1− δ)|x̃(t)|
|x̃(t)|(
2E(1 + δ))(
2E + oR→∞(1))
+O(R−1)
1 + δ
+ oR→∞(1) > 1− 2δ + oR→∞(1).(5.15)
So, if R is large enough, γ̃(t) ∈ Γ+(R/2, d, σ+), σ+ = 1− 3δ.
Then a) and b) imply that, for all γ̃ ∈ V and t > T0 := T2, we have
(5.16) γ̃(t) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 21
The lemma follows from (5.10), (5.16) and a compactness argument. �
Recall that the microsupport of g−(x)e
iψ−(x)/h ∈ C∞0 (Rn) is contained in Γ−(R−, d1, σ1).
Let ω−(x, ξ) ∈ A0 with ω− = 1 near Γ−(R−/2, d1, σ1) and supp(ω−) ⊂ Γ−(R−/3, d0, σ0).
Using the identity
(5.17) u− =
e−it(P−E)/h(g−e
iψ−/h)dt+R(E + i0, h)e−iT (P−E)/h(g−eiψ−/h),
and Proposition 4.1, Proposition 4.2 and Lemma 5.1, we get
(5.18) Oph(ω−)u− = Oph(ω−)
e−it(P−E)/h(g−e
iψ−/h)dt+O(h∞),
for some T > 0 large enough. In particular,
(5.19) MS(Oph(ω−)u−) ⊂ Λ−ω ∩ (B(0, R− + 1)× Rn).
5.2. Computation of u− along γ
Now we want to compute u− microlocally along a trajectory γ
k . We recall that γ
k is a
bicharacteristic curve (x−k (t), ξ
k (t)) such that (x
k (t), ξ
k (t)) → (0, 0) as t → +∞, and such
that, as t→ −∞,
(5.20)
|x−k (t)−
2E0ωt− z−k | → 0,
2E0ω| → 0.
If R− is large enough, a− solves (4.7) and (4.8) microlocally near γ
k ∩ MS(g−eiψ−/h). In
particular, microlocally near γ−k ∩ Γ−(R−/2, d1, σ1) ∩ (B(0, R−)× Rn), u− is given by (5.18)
e−it(P−E)/h([χ−, P ]a−e
iψ−/h)dt+O(h∞)
e−it(P−E)/h(χ−(P − E)a−eiψ−/h)dt
e−it(P−E)/h((P − E)χ−a−eiψ−/h)dt+O(h∞)
(P − E)e−it(P−E)/h(χ−a−eiψ−/h)dt+O(h∞)
=(P − E)R(E + i0, h)a−eiψ−/h +O(h∞)
iψ−/h +O(h∞).(5.21)
Now, using (5.21), and the fact that u− is a semiclassical distribution satisfying
(5.22) (P − E)u− = 0,
we can compute u− microlocally near γ
∩ B(0, R−) using Maslov’s theory (see [23] for
more details). Moreover, it is proved in Proposition C.1 (see also [5, Lemma 5.8]) that the
Lagrangian manifold Λ−ω has a nice projection with respect to x in a neighborhood of γ
close
to (0, 0). Then, in such a neighborhood, u− is given by
(5.23) u−(x) = a−(x, h)e
π/2eiψ−(x)/h,
22 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where ν−k denotes the Maslov index of γ
k , and ψ− satisfies the usual eikonal equation
(5.24) p(x,∇ψ−) = E0.
Here, to the contrary of (4.21), we have written E = E0 + zh with z = O(1), and we choose
to work with z in the amplitudes instead of the phases. As usual, we have
(5.25) ∂t(ψ−(x
(t))) = ∇ψ−(x−k (t)) · ∂tx
(t) = ∇ψ−(x−k (t)) · ξ
(t) = |ξ−
(t)|2,
so that
(5.26) ψ−(x
(t)) = ψ−(x
(s)) +
(u)|2du
We also have ψ−(x
k (s)) = (
2E0ωs+ z
k ) ·
2E0ω + o(1) as s→ −∞, and then
(5.27) ψ−(x
k (t)) = 2E0s+
|ξ−k (u)|
2du+ o(1), s→ −∞.
We have obtained in particular that
(5.28) ψ−(x
k (t)) =
|ξ−k (u)|
2−2E01u<0 du =
|ξ−k (u)|
2−V (x−k (u))+E0 sgn(u) du.
We turn to the computation of the symbol. The function a−(x, h) ∼
k=0 a−,k(x)h
satisfies the usual transport equations:
(5.29)
∇ψ− · ∇a−,0 +
(∆ψ− − 2iz)a−,0 = 0,
∇ψ− · ∇a−,k +
(∆ψ− − 2iz)a−,k = i
∆a−,k−1, k ≥ 1,
In particular, we get for the principal symbol
(5.30) ∂t(a−,0(x
(t))) = ∇a−,0(x−k (t)) · ξ
(t) = ∇a−,0(x−k (t)) · ∇ψ−(x
(t)),
so that,
(5.31) ∂t(a−,0(x
k (t))) = −
∆ψ−(x
k (t))− 2iz
a−,0(x
k (t))
and then
(5.32) a−,0(x
k (t)) = a−,0(x
k (s)) exp
∆ψ−(x−(u)) du + i(t− s)z
On the other hand, from [30, Lemma 4.3], based on Maslov theory, we have
(5.33) a−,0(x
k (t)) = (2E0)
1/4D−k (t)
−1/2eitz ,
where
(5.34) D−
(t) =
∣∣ det
∂x−(t, z, ω,E0)
∂(t, z)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 23
6. Computation of u− at the critical point
Now we use the results of [5] to get a representation of u− in a whole neighborhood of
the critical point. Indeed we saw already that (P − E)u− = 0 out of the support of g−, in
particular in a neighborhood of the critical point. First, we need to recall some terminology
of [18] and [5].
We recall from Section 2 that (µj)j≥0 is the strictly growing sequence of linear combinations
over N of the λj’s. Let u(t, x) be a function defined on [0,+∞[×U , U ⊂ Rm.
Definition 6.1. We say that u : [0,+∞[×U → R, a smooth function, is expandible, if, for
any N ∈ N, ε > 0, α, β ∈ N1+m,
(6.1) ∂αt ∂
u(t, x)−
uj(t, x)e
−µj t
e−(µN+1−ε)t
for a sequence of (uj)j smooth functions, which are polynomials in t. We shall write
u(t, x) ∼
uj(t, x)e
−µj t,
when (6.1) holds.
We say that f(t, x) = Õ(e−µt) if for all α, β ∈ N1+m and ε > 0 we have
(6.2) ∂αt ∂
xf(t, x) = O(e−(µ−ε)t).
Definition 6.2. We say that u(t, x, h), a smooth function, is of class SA,B if, for any ε > 0,
α, β ∈ N1+m,
(6.3) ∂αt ∂
xu(t, x, h) = O
hAe−(B−ε)t
Let S∞,B =
A SA,B. We say that u(t, x, h) is a classical expandible function of order (A,B),
if, for any K ∈ N,
(6.4) u(t, x, h) −
uk(t, x)h
k ∈ SK+1,B,
for a sequence of (uk)k expandible functions. We shall write
u(t, x, h) ∼
uk(t, x)h
in that case.
Since the intersection between Λ−ω and Λ− is transverse along the trajectories γ
k ), and
since g−1 (z
k ) 6= 0, Theorem 2.1 and Theorem 5.4 of [5] implies that one can write, microlocally
near (0, 0),
(6.5) u− =
∫ N−∑
αk(t, x, h)eiϕ
k(t,x)/hdt,
24 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where the αk(t, x, h)’s are classical expandible functions in S0,2ReΣ(E):
(6.6)
αk(t, x, h) ∼
αkm(t, x)h
αkm(t, x) ∼
αkm,j(t, x)e
−2(Σ(E)+µj )t,
and where the αkm,j(t, x)’s are polynomial with respect to t. We recall from (2.26) that, for
E = E0 + hz,
(6.7) Σ(E) =
− iz.
Following line by line Section 6 of [5], we obtain (see [5, (6.26)])
αk0,0(0) = e
iπ/4(2λ1)
3/2e−iν
π/2|g(γ−k )|(D
−1/2(2E0)
1/4.(6.8)
Notice that from (5.32) and Proposition C.1, we have 0 < D−
< +∞.
From [5, Section 5], we recall that the phases ϕk(t, x) satisfies the eikonal equation
(6.9) ∂tϕ
k + p(x,∇xϕk) = E0,
and that they have the asymptotic expansion
(6.10) ϕk(t, x) ∼
ϕkj,m(x)t
me−µjt,
with Mkj < +∞. In the following, we denote
(6.11) ϕkj (t, x) =
ϕkj,m(x)t
and the first ϕkj ’s are of the form
ϕk0(t, x) =ϕ+(x) + ck(6.12)
ϕk1(t, x) =− 2λ1g−(z−k ) · x+O(x
2),(6.13)
where ck ∈ R is the constant depending on k given by
(6.14) ck = “ψ−(0)” = lim
k (t)) = S
thanks to (5.28) (see also [5, Lemma 5.10]). Moreover ϕ+ is the generating function of the
outgoing stable Lagrangian manifold Λ+ with ϕ+(0) = 0. We have
(6.15) ϕ+(x) =
x2j +O(x3).
The fact that ϕk1(t, x) does not depend on t and the expression (6.13) follows also from
Corollary 6.6 and (6.109).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 25
6.1. Study of the transport equations for the phases.
Now, we examine the equations satisfied by the functions ϕkj (t, x), defined in (6.10), for
the integers j ≤ ̂ (recall that ̂ is defined by µb = 2λ1). For clearer notations, we omit the
superscript k until further notice.
Let us recall that the function ϕ(t, x) satisfies the eikonal equation (6.9), which implies
(see (6.10))
(6.16)
e−µjtϕj,m(x)(−µjtm+mtm−1)+
∇ϕj,m(x)tme−µj t
+V (x) ∼ E0,
and then
e−µjtϕj,m(x)(−µjtm +mtm−1) +
∇ϕj,m∇ϕe,em(x)e−(µj+µe)ttm+ em
+V (x) ∼ E0.(6.17)
When µj < 2λ1, the double product of the previous formula provides a term of the form e
if and only if µj = 0 or µe = 0. In particular, the term in e
−µjt in (6.17) gives
(6.18)
ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) ·
∇ϕj,m(x)tm = 0.
When µj = 2λ1, one gets also a term in e
−2λ1t for µj = µe = λ1 and then
ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) ·
∇ϕj,m(x)tm
tm+ em∇ϕ1,m(x)∇ϕ1, em(x) = 0.(6.19)
We denote
(6.20) L = ∇ϕ+(x) · ∇
the vector field that appears in (6.18) and (6.19). We set also L0 =
j λjxj∂j its linear part
at x = 0, and we begin with the study of the solution of
(6.21) (L− µ)f = g,
with µ ∈ R and f , g ∈ C∞(Rn). First of all, we show that it is sufficient to solve (6.21) for
formal series.
Proposition 6.3. Let g ∈ C∞(Rn) and g0 the the Taylor expansion of g at 0. For each
formal series f0 such that (L−µ)f0 = g0, there exists one and only one function f ∈ C∞(Rn)
defined near 0 such that f = f0 +O(x∞) and
(6.22) (L− µ)f = g,
near 0.
26 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Proof. Let f̃0 be a C
∞ function having f0 has Taylor expansion at 0. With the notation
f = f̃0 + r, the problem (6.22) is equivalent to find r = O(x∞) with
(6.23) (L− µ)r = g − (L− µ)f̃0 = r̃,
where r̃ ∈ C∞ has g0 − (L− µ)f0 = 0 as Taylor expansion at 0. Let y(t, x) be the solution of
(6.24)
∂ty(t, x) = ∇ϕ+(y(t, x)),
y(0, x) = x.
Thus, (6.23) is equivalent to
(6.25) r(x) =
e−µsr̃(y(s, x))ds + e−µtr(y(t, x)).
Since r(x), r̃(x) = O(x∞) and y(s, x) = O(eλ1t|x|) for t < 0, the functions e−µtr(y(t, x)),
e−µtr̃(y(t, x)) are O(eNt) as t→ −∞ for all N > 0. Then
(6.26) r(x) =
e−µsr̃(y(s, x))ds,
and r(x) = O(x∞). The uniqueness follows and it is enough to prove that r given by (6.26)
is C∞. We have
(6.27) ∂t(∇xy) = (∇2xϕ+(y))(∇xy),
and since ∇2xϕ+ is bounded, there exists C > 0 such that
(6.28) |∇xy(t, x)| . e−Ct,
has t → −∞. Then, e−µs(∇r̃)(y(s, x))(∂jy(t, x)) = O(eNt) as t → −∞ for all N > 0 and
∂jr(x) =
−µs(∇r̃)(y(s, x))(∂jy(t, x))ds. The derivatives of order greater than 1 can be
treated the same way. �
We denote
(6.29) Lµ = L− µ : CJxK → CJxK,
where we use the standard notation CJxK for formal series, and CpJxK for formal series of
degree ≥ p. We notice that
(6.30) Lµx
α = (L0 − µ)xα + C|α|+1JxK = (λ · α− µ)xα +C|α|+1JxK.
Recall that Iℓ(µ) has been defined in (2.20). The number of elements in Iℓ(µ) will be denoted
(6.31) nℓ(µ) = #Iℓ(µ).
One has for example n2(µ) =
n1(µ)(n1(µ)+1)
Proposition 6.4. Suppose µ ∈]0, 2λ1[. With the above notations, one has KerLµ⊕ ImLµ =
CJxK. More precisely:
i) The kernel of Lµ has dimension n1(µ), and one can find a basis (Ej1 , . . . , Ejn1(µ)
KerLµ such that Ej(x) = xj + C2JxK, j ∈ I1(µ).
ii) A formal series F = F0 +
Fjxj + C2JxK belongs to ImLµ if and only if Fj = 0 for
all j ∈ I1(µ).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 27
Remark 6.5. Thanks to Propostion 6.3, the same result is true for germs of C∞ functions
at 0. Notice that when µ 6= µj for all j, Lµ is invertible.
Proof. For a given F =
α Fαx
α ∈ CJxK, we look for solutions E =
α ∈ CJxK to the
equation
(6.32) Lµ
The calculus of the term of order x0 in (6.32) leads to the equation
(6.33) E0 = −
With this value for E0, (6.32) becomes, using again (6.30),
(6.34)
|α|=1
(λ · α− µ)Eαxα =
|α|=1
α + C2JxK.
We have two cases:
If α /∈ I1(µ), one should have
(6.35) Eα =
λ · α− µ.
If α ∈ I1(µ), the formula (6.34) becomes Fα = 0. In that case, the corresponding Eα can
be chosen arbitrarily.
Now suppose that the Eα are fixed for any |α| ≤ n− 1 (with n ≥ 2), and such that
(6.36) Lµ
|α|≤n−1
α + CnJxK.
We can write (6.32) as
(6.37) Lµ
|α|=n
α − Lµ
|α|≤n−1
+ Cn+1JxK,
or, using again (6.30),
(6.38)
|α|=n
(λ · α− µ)Eαxα =
|α|≤n
α − Lµ
|α|≤n−1
+Cn+1JxK.
Since |α| ≥ 2, one has λ · α ≥ 2λ1 > µ, so that (6.38) determines by induction all the Eα’s
for |α| = n in a unique way. �
Corollary 6.6. If j < ̂, the function ϕj(t, x) does not depend on t, i.e. we have Mj = 0.
Proof. Suppose that Mj ≥ 1, then (6.18) gives the system
(6.39)
(L− µj)ϕj,Mj = 0,
(L− µj)ϕj,Mj−1 = −Mjϕj,Mj ,
with ϕj,Mj 6= 0. But this would imply that ϕj,Mj ∈ KerLµ ∩ ImLµ, a contradiction. �
28 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
As a consequence, for j < ̂, the equation (6.18) on ϕj reduces to
(6.40) (L− µj)ϕj,0 = 0,
and, from Proposition 6.4, we get that
(6.41) ϕj(t, x) = ϕj,0(x) =
k∈I1(µ)
dj,kxk +O(x2).
Now we pass to the case j = ̂, and we study (6.19). First of all, we have seen that ϕ1 does
not depend on t, so that this equation can be written
(6.42)
ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+ ·
∇ϕj,m(x)tm +
∣∣∇ϕ1(x)
∣∣2 = 0.
As for the study of (6.18), we begin with that of (6.21), now in the case where µ = 2λ1.
We denote Ψ : Rn1(2λ1) −→ Rn2(λ1) the linear map given by
(6.43) Ψ(Eβ1 , . . . , Eβn1(2λ1)
β∈I1(2λ1)
∂α(L− µ)xβ
α∈I2(λ1)
and we set
(6.44) n(Ψ) = dimKerΨ.
Recalling that L = ∇ϕ+(x) · ∇, we see that
(6.45) Ψ(Eβ1 , . . . , Eβn1(2λ1)
β∈I1(2λ1)
∂α∂βϕ+(0)
α∈I2(λ1)
More generally, for any |α| = 2, we denote
(6.46) Ψα((Eβ)β∈I1(2λ1)) =
β∈I1(2λ1)
∂α∂βϕ+(0)
Then, at the level of formal series, we have the
Proposition 6.7. Suppose µ = 2λ1. Then
i) KerLµ has dimension n2(λ1) + n(Ψ).
ii) A formal series F =
α Fαx
α belongs to ImLµ if and only if
∀α ∈ I1(2λ1), Fα = 0,(6.47)
|β|=1
β /∈I1(2λ1)
∂β∂αϕ+(0)
2λ1 − λ · β
α∈I2(λ1)
∈ ImΨ.(6.48)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 29
iii) If F ∈ ImLµ, any formal series E =
α with LµE = F satisfies
F0,(6.49)
λ · α− 2λ1
Fα, for α ∈ I1 \ I1(2λ1),(6.50)
β∈I1(2λ1)
|β|=1
β /∈I1(2λ1)
∂β∂αϕ+(0)
2λ1 − λ · β
α∈I2(λ1)
.(6.51)
Moreover for α ∈ I2 \ I2(λ1), one has
(6.52) Eα =
λ · α− 2λ1
Fα −Ψα((Eβ)β∈I1(2λ1)) +
|β|=1
β /∈I1(2λ1)
2λ1 − λ · β
∂α+βϕ+(0)
Last, E is completely determined by F and a choice of the Eα for |α| ≤ 2 such that
(6.49)– (6.52) are satisfied.
iv) KerLµ ∩ Im(Lµ)2 = {0}.
Proof. For a given F =
α Fαx
α we look for a E =
α such that L2λ1E = F . First of
all, we must have
(6.53) E0 = −
When this is true, we get
(6.54)
|α|=1
Eα(L0 − 2λ1)xα =
|α|=1
Fα(L− 2λ1)xα + C2JxK,
and we obtain as necessary condition that Fα = 0 for any α ∈ I1(2λ1). So far, the Eα for
α ∈ I1(2λ1) can be chosen arbitrarily, and we must have
(6.55) Eα =
λ · α− 2λ1
, α ∈ I2 \ I1(2λ1).
We suppose that (6.53) and (6.55) hold. Then we should have
(6.56)
|α|=2
Eα(L0−2λ1)xα =
|α|=2
|α|=1
α/∈I1(2λ1)
|α|=1
Eα(L−2λ1)xα
+C3JxK.
Notice that the second term in the R.H.S of (6.56) belongs to C2JxK thanks to (6.55). Again,
we have to cases:
• When α ∈ I2(λ1), the corresponding Eα can be chosen arbitrarily, but one must have
|β|=1
∂α(L− 2λ1)xβ
|x=0(6.57)
=Ψα((Eβ)β∈I1(2λ1)) +
|β|=1
β /∈I1(2λ1)
∂α+βϕ+(0)
,(6.58)
30 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
and this, with (6.55), gives (6.51).
• When |α| = 2, α /∈ I2(λ1), one obtains
λ · α− 2λ1
|β|=1
∂α(L− 2λ1)xβ
λ · α− 2λ1
Fα −Ψα((Eβ)β∈I1(2λ1))−
|β|=1
β /∈I1(2λ1)
∂α+βϕ+(0)
,(6.59)
and this, with (6.55), gives (6.52).
Now suppose that (6.53), (6.55), (6.57) and (6.59) hold, and that we have chosen a value
for the free variables Eα for α ∈ I1(2λ1)∪I2(λ1). Thanks to the fact that λ ·α 6= 2λ1 for any
α ∈ Nn with |α| = 3, we see as in the proof of Propostion 6.4, that the equation (6.54) has a
unique solution, and the points (i), (ii) and (iii) follows easily.
We prove the last point of the proposition, and we suppose that
(6.60) E =
α ∈ KerLµ ∩ Im(Lµ)2.
First, we have E ∈ KerLµ ∩ ImLµ. Thus, E0 = 0 by (6.49), Eα = 0 for α ∈ I1(2λ1) by
(6.47), and Eα = 0 for α ∈ I1 \ I1(2λ1) by (6.50). Last, since LµE = 0, we also have Eα = 0
for α ∈ I2 \ I2(λ1), and finally,
(6.61) E =
α∈I2(λ1)
α + C3JxK.
Moreover, one can write E = LµG for some G ∈ ImLµ. Since E0 = 0, we must have G0 = 0.
Since G ∈ ImLµ, by (6.47), we have Gα = 0 for α ∈ I1(2λ1). Finally, since Eα = 0 for
|α| = 1, α /∈ I1(2λ1), the same is true for the corresponding Gα, and
(6.62) G =
|α|≥2
Then, since Lµx
α = 0+C3[x] for α ∈ I2(λ1), we obtain Eα = 0 for α ∈ I2(λ1). As above, we
then get that, for |α| ≥ 3, Eα = 0, and this ends the proof. �
Corollary 6.8. We always have Mb ≤ 2. If, in addition, λk 6= 2λ1 for all k ∈ {1, . . . , n}, then
Mb ≤ 1.
Proof. Suppose that Mb ≥ 3. Then (6.42) gives
(L− µb)ϕb,Mb = 0(6.63)
(L− µb)ϕb,Mb −1 = −Mbϕb,Mb(6.64)
(L− µb)ϕb,Mb −2 = −(Mb − 1)ϕb,Mb −1,(6.65)
with ϕb,Mb 6= 0. Notice that we have used the fact that Mb − 2 > 0 in (6.65). But this gives
ϕb,Mb ∈ Ker(L − µb) and (L − µb)2ϕb,Mb−2 = Mb(Mb − 1)ϕb,Mb , so that ϕb,Mb ∈ Im(L− µb)2.
This contradicts point (iv) of Proposition 6.7.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 31
Now we suppose that λk 6= 2λ1 for all k ∈ {1, . . . n}, that is I1(2λ1) = ∅, and that Mb = 2.
Then (6.42) gives
(L− µb)ϕb,Mb = 0(6.66)
(L− µb)ϕb,Mb −1 = −Mb ϕb,Mb(6.67)
with ϕb,Mb 6= 0. Therefore we have ϕb,Mb ∈ KerLµb ∩ ImLµb , and we get the same conclusion
as in (6.61): ϕb,Mb(x) = O(x2). Then, we write
(6.68) ϕb,Mb = (L− µb)g,
and we see, as in (6.62), that g = O(x2), here because I1(2λ1) = ∅. Finally, we conclude also
that ϕb,Mb = 0, a contradiction. �
6.2. Taylor expansions of ϕ+ and ϕ
Now we compute the Taylor expansions of the leading terms with respect to t, of the phase
functions ϕ(t, x) = ϕk(t, x).
Lemma 6.9. The smooth function ϕ+(x) =
x2j +O(x3) satisfies
(6.69) ∂αϕ+(0) = −
λ · α∂
αV (0),
for |α| = 3, and
(6.70) ∂αϕ+(0) = −
2(λ · α)
β,γ∈I2
α=β+γ
β! γ!
βV (0)
λj + λ · β
γV (0)
λj + λ · γ
λ · α∂
αV (0),
for |α| = 4, where α, β, γ ∈ Nn are multi-indices.
Proof. The smooth function x 7→ ϕ+(x) is defined in a neighborhood of 0, and it is charac-
terized (up to a constant: we have chosen ϕ+(0) = 0) by
(6.71)
p(x,∇ϕ+(x)) =
|∇ϕ+(x)|2 + V (x) = 0
∇ϕ+(x) = (λjxj)j=1,...,n +O(x
The Taylor expansion of ϕ+ at x = 0 is
(6.72) ϕ+(x) =
x2j +
|α|=3,4
∂αϕ+(0)
xα +O(x5),
and we have
(6.73) ∂jϕ+(x) = λjxj +
|α|=3,4
∂αϕ+(0)
xα−1j +O(x4).
32 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Therefore
|∇ϕ+(x)|2 =
j + 2
|α|=3
)∂αϕ+(0)
xα + 2
|α|=4
)∂αϕ+(0)
|α|=3
∂αϕ+(0)
xα−1j
+O(x5).(6.74)
Let us compute further the last term in (6.74):
|α|=3
∂αϕ+(0)
xα−1j
|β|,|γ|=3
∂βϕ+(0)
∂γϕ+(0)
xβ+γ−21j
|α|=4
α=β+γ
|β|,|γ|=2
βϕ+(0)
γϕ+(0)
·(6.75)
Writing the Taylor expansion of V at x = 0 as
(6.76) V (x) =
x2j +
|α|=3,4
∂αV (0)
xα +O(x5),
and using the eikonal equation (6.71), we obtain first, for any α ∈ Nn with |α| = 3,
(6.77) ∂αϕ+(0) = −
λ · α∂
αV (0).
Then, (6.74) and (6.75) give
(6.78) ∂αϕ+(0) = −
λ · α∂
αV (0) − 1
2(λ · α)
β,γ∈I2
α=β+γ
βV (0)
λj + λ · β
γV (0)
λj + λ · γ
for |α| = 4. �
Now we pass to the function ϕ1. This function is a solution, in a neighborhood of x = 0,
of the transport equation
(6.79) Lϕ1(x) = λ1ϕ1(x),
where L is given in (6.20).
Lemma 6.10. The C∞ function ϕ1(x) = −2λ1g−1 (z
) · x+O(x2) satisfies
(6.80) ∂αϕ1(0) =
2λ1α!
(λ1 − λ · α)(λ1 + λ · α)
αV (0)
g−1 (z
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 33
for |α| = 2, and
∂αϕ1(0) =−
λ1 − λ · α
k∈I1(λ1),j∈I1
β,γ∈I2
α+1j=β+γ
βV (0)
λj + λ · β
γV (0)
(λ1 − λ · γ)(λ1 + λ · γ)
g−1 (z
(λ1 − λ · α)(λ1 + λ · α)
k∈I1,j∈I1(λ1)
β,γ∈I2
1j+α=β+γ
(α+ 1j)!
βV (0)
λk + λ · β
γV (0)
λk + λ · γ
g−1 (z
(λ1 − λ · α)(λ1 + λ · α)
j∈I1(λ1)
αV (0)
g−1 (z
.(6.81)
for |α| = 3.
Proof. We write
(6.82) ϕ1(x) =
ajxj +
|α|=2,3
α +O(x4),
and Lemma 6.9 together with (6.73) give all the coefficients in the expansion
(6.83) ∇ϕ+(x) =
λjxj +
|α|=2,3
Aj,αx
α +O(x4)
j=1,...,n
In fact, we have
(6.84) Aj,α =
∂α+1jϕ+(0)
and aα =
∂αϕ1(0)
We get
Lϕ1(x) =
∂jϕ+(x)∂jϕ1(x)
ajλjxj +
|α|=2
αjλjaα + ajAj,α
|α|=3
αjλjaαx
|β|=|γ|=2
Aj,βγjaγx
β+γ−1j +
|α|=3
ajAj,αx
+O(x4)
ajλjxj +
|α|=2
λ · α aα +
Aj,αaj
|α|=3
λ · α aα +
α=β+γ−1j
|β|,|γ|=2
Aj,βγjaγ + ajAj,α
xα +O(x4).(6.85)
34 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Thus, (6.79) gives, for all α ∈ Nn with |α| = 2,
(6.86) aα =
λ1 − λ · α
Aj,αaj ,
and, for all α ∈ Nn with |α| = 3,
(6.87) aα =
λ1 − λ · α
β,γ∈I2
α+1j=β+γ
γjAj,βaγ + ajAj,α
Then, the lemma follows from (6.84). �
6.3. Asymptotics near the critical point for the trajectories.
The informations obtained so far are not sufficient for the computation of the ϕj ’s. We
shall obtain here some more knowledge by studying the behaviour of the incoming trajectory
γ−(t) as t → +∞. We recall from [18, Section 3] (see also [5, Section 5]), that the curve
γ−(t) = (x−(t), ξ−(t)) ∈ Λ− ∩ Λ−ω satisfy, in the sense of expandible functions,
(6.88) γ−(t) =
M ′j∑
γ−j,mt
me−µjt,
Notice that we continue to omit the subscript k for γ−k = (x
k , ξ
k ), z
k , . . . Writing also
(6.89) x−(t) ∼
g−j,m(t, z−)e
−µj t, g−j (z
−, t) =
M ′j∑
g−j,m(z
−)tm,
for some integers M ′j, we know that g
−) = g−1,0(z
−) 6= 0. Since ξ−(t) = ∂tx−(t), we have
(6.90) ξ−(t) ∼
M ′j∑
g−j,m(z
−)(−µjtm +mtm−1)e−µjt.
Proposition 6.11. If j < ̂, then M ′j = 0. We also have M
≤ 1, and M ′
= 0 when
I1(2λ1) 6= ∅. Moreover
(6.91) (g−
|α|=2
∂α+βV (0)
(g−1 (z
−))α for β ∈ I1(2λ1),
0 for β /∈ I1(2λ1).
and, for |β| = 1, β /∈ I1(2λ1),
(6.92) (g−
(2λ1 + λ · β)(2λ1 − λ · β)
|α|=2
∂α+βV (0)
(g−1 (z
−))α.
Proof. First of all, since ∂tγ
−(t) = Hp(γ
−(t)), we can write
(6.93) ∂tγ
−(t) = Fp(γ
−(t)) +O(t2M ′1e−2λ1t),
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 35
where
(6.94) Fp = d(0,0)Hp =
, Λ2 = diag(λ21, . . . , λ
We obtain
(6.95)
1≤j<b
M ′j∑
(Fp + µj)γ
1≤j<b
M ′j∑
γ−j,mmt
m−1e−µj t.
Now suppose j < ̂ and M ′j ≥ 1. We get, for this j, for some γ
j,M ′j
6= 0,
(6.96)
(Fp + µj)γ
j,M ′j
(Fp + µj)γ
j,M ′j−1
=M ′jγ
j,M ′j
so that Ker(Fp + µj) ∩ Im(Fp + µj) 6= {0}. Since Fp is a diagonizable matrix, this can easily
be seen to be a contradiction.
Now we pass to the study of M ′
. So far we have obtained that
(6.97) γ−(t) =
1≤j<b
γ−j e
−µjt +
tme−2λ1t +O(tCe−µb+1t),
and we can write
(6.98) Hp(x, ξ) =
|α|=2
∂α∇V (0)
xα +O(x3)
 .
Thus we have
(6.99) Hp(γ
−(t)) = Fp
γ−j e
−µjt +
tme−2λ1t
+ e−2λ1tA(γ−1 ) +O(e
−(2λ1+ε)t),
where, noticing that µj + µj′ = 2λ1 if and only if j = j
′ = 1,
(6.100) A(γ−1 ) =
|α|=2
∂α∇V (0)
(g−1 )
 .
For the terms of order e−2λ1t, we have, since ∂tγ
−(t) = Hp(γ
−(t)),
(6.101) (Fp + 2λ1)
mtm−1 −A(γ−1 ).
Thus, if we suppose that M ′
≥ 2, we obtain
(6.102)
(Fp + 2λ1)γ
b,M ′
(Fp + 2λ1)γ
b,M ′
b,M ′
36 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Then again we have γ−
b,M ′
∈ Ker(Fp + 2λ1) ∩ Im(Fp + 2λ1), a contradiction.
Eventually, if λj 6= 2λ1 for all j, then Ker(Fp + 2λ1) = {0}. Therefore, if we suppose that
b = 1, we see that γb,1 6= 0 satisfies the first equation in (6.102) and we get a contradiction.
Now we compute γ−
(t) = γ−
t+ γ−
. We have
(6.103)
(Fp + 2λ1)γ
(Fp + 2λ1)γ
−A(γ−1 ),
and we see that γ−
= Πγ−
= ΠA(γ−1 ), where Π is the projection on the eigenspace of
Fp associated to −2λ1. We denote by ej = (δi,j ⊗ 0)i=1,...,n and εj = (0 ⊗ δi,j)i=1,...,n for
j = 1, . . . , n, so that (e1, . . . en, ε1, . . . , εn) is the canonical basis of R
2n = T(0,0)T
n. Then it
is easy to check that , for all j, v±j = ej±λj1εj is an eigenvector of Fp for the eigenvalue ±λj .
In the basis {e1, ε1, . . . , en, εn} the projector Π is block diagonal and, if Kj = Vect(ej , εj), we
(6.104) Π|Kj
1/2 −1/4λ1
−λ1 1/2
for j ∈ I1(2λ1),
0 for j /∈ I1(2λ1).
Therefore, we obtain
(6.105) (g−
|α|=2
∂β∂αV (0)
(g−1 (z
−))α for β ∈ I1(2λ1),
0 for β /∈ I1(2λ1).
Now suppose that k /∈ I1(2λ1). Then the second equality in (6.103) restricted to Kk gives
(6.106)
2λ1 1
λ2k 2λ1
Πkγb,0 = −ΠkA(γ−1 ),
where Πk denotes the projection onto Kk. Solving this system, one gets
(6.107) (g−
4λ21 − λ2k
ΠxΠkA(γ
and, together with (6.100), this ends the proof of Proposition 6.11. �
6.4. Computation of the ϕkj ’s.
Here we compute the ϕkj ’s for j ≤ ̂. We still omit the superscript k. From [5], we know
that ξ−(t) = ∇xϕ
t, x−(t)
, so that, using (6.41),
ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t +
2≤j<b
∇ϕj(0)e−µj t
+∇ϕb,2(0)t2e−2λ1t +∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t).(6.108)
Since ϕ+ = −ϕ− and ξ− ∈ Λ−, we have ∇ϕ+(x−(t)) = −ξ−(t), and we obtain first, by (6.90),
(6.109) ∇ϕj(0) = −2µjg−j (z
for 1 ≤ j < ̂.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 37
Now we study ϕb(t, x) = ϕb,0(x) + tϕb,1(x) + t
2ϕb,2(x) when I1(2λ1) 6= ∅. It follows from
(6.108) that we have
(6.110)
− 4λ1g−b,1(z
−) = ∇ϕb,1(0),
− 4λ1g−b,0(z
−) + 2g−
(z−) = ∇ϕb,0(0) +∇2ϕ1(0)g−1 (z
On the other hand, we have seen that, by (6.19), the functions ϕb,2, ϕb,1 and ϕb,0 satisfy
(6.111)
(L− 2λ1)ϕb,2 = 0,
(L− 2λ1)ϕb,1 = −2ϕb,2,
(L− 2λ1)ϕb,2 = −ϕb,1 −
|∇ϕ1(0)|2.
In particular ϕb,2 ∈ Ker(L− 2λ1) ∩ Im(L− 2λ1) so that (see (6.61)),
(6.112) ϕb,2(x) =
α∈I2(λ1)
c2,αx
α +O(x3).
Going back to (6.108), we notice that we obtain now
ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t +
2≤j<b
∇ϕj(0)e−µj t
∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t),(6.113)
and this equality is consistent with Proposition 6.11.
Then, (6.49) and (6.50) give
(6.114) ϕb,1(x) =
α∈I1(2λ1)
c1,αx
|α|=2
c1,αx
α +O(x3),
and, by (6.51), we have
(6.115) Ψ((c1,β)β∈I1(2λ1)) = (−2c2,α)α∈I2(λ1).
By (6.52), we also have for |α| = 2, α /∈ I2(λ1),
(6.116) c1,α =
2λ1 − λ · α
β∈I1(2λ1)
∂α+βϕ+(0)
c1,β .
The function ϕb,0(x) =
|α|≤2 c0,αx
α +O(x3) satisfies (see (6.42))
(6.117) (L− 2λ1)ϕb,0 = −ϕb,1 −
∣∣∇ϕ1(x)
First of all, the compatibility condition (6.47) gives
(6.118) ∀α ∈ I1(2λ1), c1,α = −∇ϕ1(0) · ∂α∇ϕ1(0),
so that in particular, by (6.115), the function ϕb,2 is known up to O(x3) terms:
(6.119) ∀α ∈ I2(λ1), c2,α =
β∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0),
38 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
(6.120) ∀α /∈ I2(λ1), |α| = 2, c1,α = −
2λ1 − λ · α
β∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0).
Now (6.49) and (6.50) give
(6.121) c0,0 = ϕb,0(0) =
|∇ϕ1(0)|2,
(6.122) ∀α /∈ I1(2λ1), |α| = 1, c0,α =
2λ1 − λ · α
∇ϕ1(0) · ∂α∇ϕ1(0).
From the other compatibility condition (6.48), we know that
c1,α +
∇ϕ1(0) · ∂α∇ϕ1(0) +
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0)
|β|=1
β/∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0)
2λ1 − λ · β
α∈I2(λ1)
∈ ImΨ,(6.123)
and, from (6.51), we obtain a relation between the (c0,β)β∈I1(2λ1) and the (c1,α)α∈I2(λ1),
namely
∀α ∈ I2(λ1), c1,α =−
∂α∇ϕ1(0) · ∇ϕ1(0)−
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0)
β∈I1(2λ1)
∂α+βϕ+(0)
c0,β −
|β|=1
β/∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0)
2λ1 − λ · β
·(6.124)
Using the second equation in (6.110), we obtain, for |β| = 1,
(6.125) c0,β = −4λ1(g−b,0(z
−))β + 2(g−
(z−))β − ∂β∇ϕ1(0) · g−1 (z
At this point, we have computed the functions ϕb,1(x) and ϕb,2(x) up to O(x3), in terms
of derivatives of ϕ+ and ϕ1, and of the g
(z−). We shall now use the expressions we have
obtained in Section 6.2 and in Section 6.3 to give these functions in terms of g−1 and of
derivatives of V only.
First of all, by (6.112), (6.119), Lemma (6.9) and Lemma (6.10), we obtain
ϕb,2(x) =−
γ∈I1(2λ1)
α,β∈I2(λ1)
∂β+γV (0)
(g−1 (z
∂α+γV (0)
+O(x3).(6.126)
Then we have
(6.127) ϕb,1(x) = −4λ1g−b,1(z
−) · x+
α∈I2(λ1)
c1,αx
|α|=2
α/∈I2(λ1)
c1,αx
α +O(x3),
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 39
where the c1,α are given by (6.124) and (6.125) for α ∈ I2(λ1), and by (6.120) for α /∈ I2(λ1).
• For |α| = 2, α /∈ I2(λ1), we obtain by (6.116), Lemma 6.9 and Lemma 6.10,
c1,α =
(2λ1 + λ · α)(2λ1 − λ · α)
β∈I1(2λ1)
∂α+βV (0)
(λ1 + λj)(3λ1 + λj)
β∇V (0) · g−1 (z−)(g
−))j .(6.128)
Since (g−1 (z
−))j = 0 but for j ∈ I1(λ1), we get, changing notations a bit,
(6.129) c1,α =
(2λ1 + λ · α)(2λ1 − λ · α)
γ∈I1(2λ1)
β∈I2(λ1)
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))β.
• Now we compute c1,α for α ∈ I2(λ1).
For the last term in the R.H.S. of (6.124), we obtain
|β|=1
β /∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0)
2λ1 − λ · β
γ∈I1\I1(2λ1)
β∈I2(λ1)
(2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))β .(6.130)
Using (6.91) and (6.125), we have also
β∈I1(2λ1)
∂α+βϕ+(0)
c0,β =
γ∈I1(2λ1)
∂α+γV (0)
(z−))γ +
γ∈I1(2λ1)
β∈I2(λ1)
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))β .(6.131)
40 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
We pass to the computation of − 1
∂α∇ϕ1(0) · ∇ϕ1(0) for α ∈ I2(λ1). We obtain
∂α∇ϕ1(0) · ∇ϕ1(0) = −
β∈I2(λ1)
∂α+βV (0)
(g−1 (z
j,p,k=1
β,γ∈I2
β+γ=α+1p+1j
((α+ 1p)j + 1)(αp + 1)
(λk + λ · β)(λk + λ · γ)
∂β+1kV (0)
∂γ+1kV (0)
(g−1 (z
−))j(g
+ 2λ1
j,p,k=1
β,γ∈I2
β+γ=α+1p+1j
(αp + 1)γj
(λ1 − λ · γ)(λ1 + λ · γ)(λj + λ · β)
β+1jV (0)
∂γ+1kV (0)
(g−1 (z
−))k(g
= I + II + III.
(6.132)
Writing δ = 1j + 1p, we get
(6.133) II = −1
β,γ,δ∈I2
β+γ=α+δ
(α+ δ)!
(λk + λ · β)(λk + λ · γ)
∂β+1kV (0)
∂γ+1kV (0)
(g−1 (z
α! δ!
Since δ ∈ I2(λ1) (otherwise (g−1 (z−))δ = 0), we have β, γ ∈ I2(λ1) and, changing notations a
(6.134) II = −1
β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
In the last term III, we can suppose that γ = 1j+1q for some q ∈ {1, . . . , n}. Then γj = γ!
and, writing β = 1a + 1b we have
III = λ1
j,k,p=1
(αp + 1)(g
−))k(g
a,b,q∈I1
1a+1b+1q=α+1p
(αp + 1)
(λ1 − λj − λq)(λ1 + λj + λq)(λj + λa + λb)
∂j,a,bV (0)∂j,q,kV (0).(6.135)
Since α ∈ I2(λ1) and 1p ∈ I1(λ1) (otherwise (g−1 (z−))p = 0), we have 1a, 1b, 1q ∈ I1(λ1) so
that we can write
(6.136)
III = −
j,k,p=1
(αp + 1)
λj(2λ1 + λj)2
(g−1 (z
−))k(g
a,b,q∈I1
1a+1b+1q=α+1p
∂j,a,bV (0)∂j,q,kV (0).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 41
Now it is easy to check, noticing that (α+ 1p)k ∈ {1, 2, 3, 4} and examining each case, that
(6.137)
a,b,q∈I1
1a+1b+1q=α+1p
∂j,a,bV (0)∂j,q,kV (0) =
(α+ 1p)k
a,b,c,d∈I1
1a+1b+1c+1d=α+1p+1k
∂j,a,bV (0)∂j,c,dV (0).
Therefore, we have
III = −1
j,k,p=1
(α + 1p + 1k)!
λj(2λ1 + λj)2
(g−1 (z
−))k(g
a,b,c,d∈I1
1a+1b+1c+1d=α+1p+1k
∂j,a,bV (0)∂j,c,dV (0).(6.138)
Eventually, setting β = 1p + 1k, γ = 1a + 1b and δ = 1c + 1d, we get
(6.139)
III = −
β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
λj(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
We are left with the computation of
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0) = −
β,γ∈I1(λ1)
β+γ=α
βϕ1(0) · ∂j∂γϕ1(0)
λ2j(2λ1 + λj)
β,γ∈I1(λ1)
β+γ=α
k,ℓ=1
∂j∂k∂
βV (0)(g−1 (z
−))k∂j∂ℓ∂
γV (0)(g−1 (z
−))ℓ.(6.140)
At this point, we notice that
α∈I2(λ1)
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0)xα
λ2j (2λ1 + λj)
β,γ∈I1(λ1)
α∈I2(λ1)
β+γ=α
k,ℓ=1
∂j∂k∂
βV (0)(g−1 (z
−))k∂j∂ℓ∂
γV (0)(g−1 (z
−))ℓ x
λ2j (2λ1 + λj)
α,β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
αV (0)
βV (0)
xα(g−1 (z
}(6.141)
42 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
From (6.124), (6.130), (6.131) (6.139), and (6.141), we finally obtain that
α∈I2(λ1)
c1,αx
γ∈I1\I1(2λ1)
α,β∈I2(λ1)
(2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))βxα
γ∈I1(2λ1)
α∈I2(λ1)
∂α+γV (0)
(z−))γxα +
γ∈I1(2λ1)
α,β∈I2(λ1)
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))βxα
α,β∈I2(λ1)
∂α+βV (0)
(g−1 (z
−))βxα
α,β∈I2(λ1)
(α+ β)!
γ,δ∈I2
γ+δ=α+β
(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
λj(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
(α + β)!
γ,δ∈I2(λ1)
γ+δ=α+β
λ2j(2λ1 + λj)
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
λ2j(2λ1 + λj)
αV (0)
βV (0)
xα(g−1 (z
−))β ,
(6.142)
or, more simply,
α∈I2(λ1)
c1,αx
α = −
γ∈I1(2λ1)
α∈I2(λ1)
∂α+γV (0)
(z−))γxα +
α,β∈I2(λ1)
(g−1 (z
γ∈I1\I1(2λ1)
(2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2
∂α+γV (0)∂β+γV (0)
γ∈I1(2λ1)
∂α+γV (0)∂β+γV (0)− ∂α+βV (0)
− (α+ β)!
γ,δ∈I2
γ+δ=α+β
γV (0)
δV (0)
λ2j (2λ1 + λj)
αV (0)∂j∂
βV (0)
.(6.143)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 43
7. Computations after the critical point
7.1. Stationary phase expansion in the outgoing region.
Now we compute the scattering amplitude starting from (4.19). First of all, we change the
cut-off function χ+ so that the support of the right hand side of the scalar product in (4.19)
is close to (0, 0).
ℓχ+ = 0
supp(∇χ+)
χ+ = 1
ℓeχ+ = 0
supp(∇eχ+)
eχ+ = 1
Figure 1. The support of χ+ and χ̃+ in T
Using Maslov’s theory, we construct a function v+ which coincides with a+(x, h)e
iψ+(x)/h
out of a small neighborhood of
∩ (B(0, R+ +1)×Rn) and such that v+ is a solution of
(P−E)v+ = 0 microlocally near
. Let χ̃+(x, ξ) ∈ C∞(T ∗Rn) such that χ̃+(x, ξ) = χ+(x)
out of a small enough neighborhood of
∩ (B(0, R++1)×Rn). In particular, (P −E)v+
is microlocally 0 near the support of χ+ − χ̃+. So, we have
〈u−, [χ+, P ]v+〉 =〈u−, [Op(χ̃+), P ]v+〉+ 〈u−, (χ+ −Op(χ̃+))(P − E)v+〉
− 〈(P − E)u−, (χ+ −Op(χ̃+))v+〉
=〈u−, [Op(χ̃+), P ]v+〉+O(h∞)− 〈g−eiψ−/h, (χ+ −Op(χ̃+))v+〉
=〈u−, [Op(χ̃+), P ]v+〉+O(h∞),(7.1)
since the microsupport of g−e
iψ−/h and χ+− χ̃+ are disjoint. Thus, the scattering amplitude
is given by
(7.2) A(ω, θ,E, h) = c(E)h−(n+1)/2〈u−, [Op(χ̃+), P ]v+〉+O(h∞).
Now we will prove that, modulo O(h∞), the only contribution to the scattering amplitude
in (7.2) comes from the values of the functions u− and v+ microlocally on the trajectories γ
and γ∞j . From (5.18), the fact that u− = O(h−C) and (P −E)u− = 0 microlocally out of the
microsupport of g−e
−iψ−/h, and the usual propagation of singularities theorem, we get
(7.3) MS(u−) ⊂ Λ−ω ∪ Λ+.
Moreover, we have
(7.4) MS(v+) ⊂ Λ+θ .
44 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Now, let f∞j (resp. f
ℓ ) be C
n) functions with support in a small enough neighborhood
of γ∞j (resp. γ
ℓ ∩MS(v+)) such that f∞j = 1 (resp. f
k = 1) in a neighborhood of γ
j (resp.
γ+ℓ ∩MS(v+)). In particular, we assume that all these functions have disjoint support. Since
u− and v+ have disjoint microsupport out of the support of the f
j and the f
ℓ , we have
A(ω, θ,E, h) =c(E)h−(n+1)/2
〈Op(f∞j )u−,Op(f∞j )[Op(χ̃+), P ]v+〉
+ c(E)h−(n+1)/2
〈Op(f+ℓ )u−,Op(f
ℓ )[Op(χ̃+), P ]v+〉+O(h
=Areg +Asing.(7.5)
Concerning the terms which contain f∞j , Areg, we are exactly in the same setting as in [30,
Section 4]. The computation there gives
(7.6) Areg =
j,m(ω, θ,E)h
iS∞j /h +O(h∞).
Now we compute Asing. Proceeding as in Section 5.2 for u−, one can show that v+ can be
written as
(7.7) v+(x) = a+(x, h)e
π/2eiψ+(x)/h,
microlocally near any ρ ∈ γ+
close enough to (0, 0). Here ν+
is the Maslov index of γ+
. The
phase ψ+ and the classical symbol a+ satisfy the usual eikonal and transport equations. In
particular, as in (5.28) and (5.33), we have
(7.8)
ℓ (t)) = −
|ξ+ℓ (u)|
2 − 2E01u>0 du = −
|ξ+ℓ (u)|
2 − V (x+ℓ (u)) −E0 sgn(u) du,
and a+(x, h) ∼
m a+,m(x)h
m with
(7.9) a+,0(x
ℓ (t)) = (2E0)
1/4(D+ℓ (t))
−1/2eitz,
where
(7.10) D+ℓ (t) =
∣∣ det
∂x+(t, z, θ, E0)
∂(t, z)
We can chose χ̃+ so that the microsupport of the symbol of Op(f
ℓ )[Op(χ̃+), P ] is contained
in a vicinity of such a point ρ ∈ γ+ℓ (see Figure 1). Then, microlocally near ρ, we have
(7.11) Op(f+
)[Op(χ̃+), P ]v+ = ã+(x, h)e
π/2eiψ+(x)/h,
(7.12) ã+(x, h) =
ã+,m(x)h
(7.13) ã+,0(x) = −i{χ̃+, p}(x,∇ψ+(x))a+,0(x).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 45
From [5, Section 5], the Lagrangian manifold
{(x,∇xϕk(t, x)); ∂tϕk(t, x) = 0},
coincides with Λ−ω . In particular, since MS(v+) ⊂ Λ+θ and since there is no curve γ∞(z∞j )
sufficiently closed to the critical point, the finite times in (6.5) give a contribution O(h∞)
to the scattering amplitude (4.19). In view of the equations (6.5), (6.12) and (7.11), the
principal contribution of Asing will come from the intersection of the manifolds Λ+θ and Λ+.
Recall that, from (A5), the manifolds Λ+θ and Λ+ intersect transversally along γ
In particular, to compute Asing, we can apply the method of stationary phase in the
directions that are transverse to γ+
. For each ℓ, after a linear and orthonormal change of
variables, we can assume that g+
) is collinear to the xℓℓℓ–direction, and that V (x) satisfies
(A2). We denote Hℓxℓℓℓ = {y = (y1, . . . , yn) ∈ R
n; yℓℓℓ = xℓℓℓ} the hyperplane orthogonal to
(0, . . . , 0, xℓℓℓ, 0, . . . , 0).
We shall compute Asing in the case where there is only one incoming curve γ−
and one
outgoing curve γ+
. In the general case, Asing is simply given by the sum over k and ℓ of such
contributions. Using (4.19), (6.5) and (7.11), we can write
Asing =c(E)h
−(n+1)/2
k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e
π/2dt dx
c(E)h−(n+1)/2√
y∈Hxℓℓℓ
k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e
π/2dt dy dxℓℓℓ.(7.14)
Let Φ(y) = ϕk(t, xℓℓℓ, y) − ψ+(xℓℓℓ, y) be the phase function in (7.14). From (6.10)–(6.13), we
can write
(7.15) Φ(y) = S−k + (ϕ+ − ψ+)(xℓℓℓ, y) + ψ̃(t, xℓℓℓ, y),
where ψ̃ = O(e−λ1t) is an expandible function. Since the manifolds Λ+
and Λ+ intersect
transversally along γ+
, the phase function y → (ϕ+−ψ+)(xℓℓℓ, y) has a non degenerate critical
point yℓ(xℓℓℓ) ∈ Hℓxℓℓℓ ∩ Πxγ
, and xℓℓℓ 7→ yℓ(xℓℓℓ) is C∞ for xℓℓℓ 6= 0. Then, from the implicit
function theorem, the function Φ has a unique critical point yℓ(t, xℓℓℓ) ∈ Hℓxℓℓℓ for t large enough
depending on xℓℓℓ. The function (t, xℓℓℓ) 7→ yℓ(t, xℓℓℓ) is expandible and we have
(7.16) yℓ(t, xℓℓℓ) = y
ℓ(xℓℓℓ)−Hess(ϕ+ − ψ+)−1
yℓ(xℓℓℓ)
yℓ(xℓℓℓ)
e−µ1t + Õ
e−µ2t
As a consequence, Φ
yℓ(t, xℓℓℓ)
is also expandible.
Since ϕ+ and ψ+ satisfy the same eikonal equation, we get (see (5.25))
(7.17) ∂t(ϕ+ − ψ+)(x+ℓ (t)) = |ξ
(t)|2 − |ξ+
(t)|2 = 0.
46 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Thus, (ϕ+ − ψ+)(yℓ(xℓℓℓ)) does not depend of xℓℓℓ and is equal to
(ϕ+ − ψ+)(yℓ(xℓℓℓ)) = lim
(ϕ+ − ψ+)(x+ℓ (t))
|ξ+ℓ (s)|
2 − 2E01s>0 ds
(s)|2 − V (x+
(s))− E0 sgn(s) ds
,(7.18)
where we have used (7.8). Therefore, the phase function Φ at the critical point yℓ(t, xℓℓℓ) is
equal to
yℓ(t, xℓℓℓ)
µm≤2λ1
t, yℓ(xℓℓℓ)
e−µmt
Hess(ϕ+ − ψ+)−1
yℓ(xℓℓℓ)
yℓ(xℓℓℓ)
· ∇ϕ1
yℓ(xℓℓℓ)
e−2µ1t + Õ(e−eµt),(7.19)
where µ̃ is the first of the µj’s such that µj > 2λ1.
Using the method of the stationary phase for the integration with respect to y ∈ Hℓxℓℓℓ in
(7.14), we get
(7.20) Asing = c(E)h
−(n+1)/2
(2πh)(n−1)/2
eiΦ(y
ℓ(t,xℓℓℓ))/hf ℓ(t, xℓℓℓ, h) dt dxℓℓℓ +O(h∞).
TheO(h∞) term follows from the fact that the error term stemming from the stationary phase
method can be integrated with respect to time t, since αk ∈ S0,2ReΣ(E), with ReΣ(E) > 0
(see the beginning of Section 6). The symbol f ℓ(t, xℓℓℓ, h) is a classical expandible function of
order S1,2ReΣ(E) in the sense of Definition 6.2:
(7.21) f ℓ(t, xℓℓℓ, h) ∼
f ℓm(t, xℓℓℓ, lnh)h
where the f ℓm are polynomials with respect to lnh and
(7.22) f ℓ0(t, xℓℓℓ, lnh) = α
t, yℓ(t, xℓℓℓ)
ã+,0
yℓ(t, xℓℓℓ)
π/2 e
i sgnΦ′′
|Hℓxℓℓℓ
(yℓ(t,xℓℓℓ))π/4
∣∣detΦ′′|Hℓxℓℓℓ
yℓ(t, xℓℓℓ)
)∣∣1/2
Using Proposition C.1, we compute the Hessian of Φ, and we get
yℓ(xℓℓℓ)
=diag(−λ1, . . . ,−λℓℓℓ−1, λℓℓℓ,−λℓℓℓ+1, . . . ,−λn) + o(1),
yℓ(xℓℓℓ)
=diag(λ1, . . . , λn) + o(1).
Then, for xℓℓℓ small enough and t large enough depending on xℓℓℓ, we have
∣∣detΦ′′|Hℓxℓℓℓ
yℓ(t, xℓℓℓ)
)∣∣1/2 =
j 6=ℓℓℓ
2λj + o(1),(7.23)
sgnΦ′′|Hℓxℓℓℓ
yℓ(t, xℓℓℓ)
= n− 1,(7.24)
as xℓℓℓ goes to 0.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 47
7.2. Behaviour of the phase function Φ.
Suppose that j ∈ N is such that j < ̂. From (6.40), we have
(7.25) ϕkj (x
(s0)) = e
−µj(s−s0)ϕkj (x
(s)).
Combining (6.41) with (6.109), we get
ϕkj (x
ℓ (s0)) =e
µjs0e−µjs
− 2µj〈g−j (z
k )|g
ℓ 〉)e
µjs +O(e2λ1s)
=− 2µj
g−j (z
∣∣g+j (z
eµjs0 .(7.26)
We suppose first that we are in the case (a) of assumption (A7). Then, (7.19) becomes
(7.27) Φ
yℓ(t, xℓℓℓ)
− 2µk
eµks(xℓℓℓ)e−µkt + Õ(e−µk+1t).
Here s(xℓℓℓ) is such that x
(s(xℓℓℓ)) = x
ℓ(xℓℓℓ) and the Õ(e−µk+1t) is in fact expandible, uniformly
with respect to xℓℓℓ when xℓℓℓ varies in a compact set avoiding 0.
Suppose now that we are in the case (b) of assumption (A7). Of course, from (7.26), we
have ϕj
yℓ(xℓℓℓ)
= 0 for all j < ̂. On the other hand, Corollary 6.8 and (6.111) imply
(7.28) ϕk
b,2(x
ℓ (s0)) = e
−2λ1(s−s0)ϕk
b,2(x
ℓ (s)).
Combining this with (6.126), we get
b,2(x
(s0)) =e
2λ1s0e−2λ1s
j∈I1(2λ1)
α,β∈I2(λ1)
∂α+1jV (0)
∂β+1jV (0)
g−1 (z
g+1 (z
e2λ1s
+O(e3λ1s)
j∈I1(2λ1)
α,β∈I2(λ1)
∂α+1jV (0)
∂β+1jV (0)
g−1 (z
g+1 (z
e2λ1s0 .(7.29)
In particular, (7.19) becomes, in that case,
yℓ(t, xℓℓℓ)
=S−k − S
j∈I1(2λ1)
α,β∈I2(λ1)
∂α+1jV (0)
∂β+1jV (0)
g−1 (z
g+1 (z
e2λ1s(xℓℓℓ)
× t2e−2λ1t +O(te−2λ1t)
=S−k + S
ℓ +M2(k, ℓ)t
2e−2λ1t +O(te−2λ1t).(7.30)
As in (7.27), the term O(te−2λ1t) is in fact expandible uniformly with respect to xℓℓℓ when xℓℓℓ
varies in a compact set avoiding 0.
Eventually, we suppose that we are in the case (c) of assumption (A7). Then we obtain
from (7.26) and (7.29) that ϕj
yℓ(xℓℓℓ)
= 0 for all j < ̂ and ϕb,2
yℓ(xℓℓℓ)
= 0. With the last
identity in mind, Equation (6.111) on ϕk
implies
(7.31) ϕk
b,1(x
ℓ (s0)) = e
−2λ1(s−s0)ϕk
b,1(x
ℓ (s)).
48 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
In order to compute ϕk
b,1(x
ℓ (s)), we put the expansion (2.15) for x
ℓ (s) (with Proposition 6.11
in mind) into (6.127). The third term in (6.127) will be, at least, O(e(µ2+µ1)s) = o(e2λ1s).
Thank to (6.91) and thanks to the fact that M2(k, ℓ) = 0, the first term in (6.127) will give
no contribution of order se2λ1s and will be of the form
(7.32) − 4λ1g−b,1(z
−) · x+ℓ (s) = −
α∈I2(λ1)
αV (0)
(g−1 (z
−))α(g+
(z+))je
2λ1s + Õ(eµb+1s)
It remains to study the contribution the second term in (6.127), as given in (6.143). As
previously, the first term of the third line in (6.143) will give a term of order o(e2λ1s). The
other terms will contribute to the order e2λ1s for
α∈I2(λ1)
αV (0)
(z−))j(g
+))α +
α,β∈I2(λ1)
(g−1 (z
(g+1 (z
− ∂α+βV (0) +
j∈I1\I1(2λ1)
λ2j (4λ
1 − λ2j )
∂α+γV (0)∂β+γV (0)
γ,δ∈I2(λ1)
γ+δ=α+β
(γ + δ)!
γ! δ!
γV (0)∂j∂
δV (0)
.(7.33)
Thus, combining (7.32) and (7.33), the discussion above leads to
b,1(x
ℓ (s0)) =e
2λ1s0e−2λ1s
M1(k, ℓ)e2λ1s + o(e2λ1s)
=M1(k, ℓ)e2λ1s0 .(7.34)
In particular, (7.19) becomes, in that case,
yℓ(t, xℓℓℓ)
=S−k + S
ℓ +M1(k, ℓ)e
2λ1s(xℓℓℓ)te−2λ1t +O(e−2λ1t).(7.35)
As above, the O(e−2λ1t) is expandible uniformly with respect to the variable xℓℓℓ when xℓℓℓ varies
in a compact set avoiding 0.
7.3. Integration with respect to time.
Now we perform the integration with respect to time t in (7.20). We follow the ideas of
[18, Section 5] and [5, Section 6]. Since yℓ(t, xℓℓℓ) is expandible (see (7.16)), and since Φ is C
outside of xℓℓℓ = 0, the symbol f
ℓ(t, xℓℓℓ, h) is expandible.
We compute only the contribution of the principal symbol (with respect to h) of f ℓ, since
the other terms can be treated the same way, and the remainder term will give a contribution
O(h∞) to the scattering amplitude. In other word, we compute
(7.36) Asing0 =
c(E)h−(n+1)/2√
(2πh)(n−1)/2h
eiΦ(y
ℓ(t,xℓℓℓ))/hf ℓ0(t, xℓℓℓ) dt dxℓℓℓ +O(h∞).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 49
First, we assume that we are in the case (a) of the assumption (A7). In that case, Ψ is
given by (7.27). For xℓℓℓ fixed in a compact set outside from 0, we set
yℓ(t, xℓℓℓ)
− (S−k + S
=− 2µk
eµks(xℓℓℓ)e−µkt +R(t, xℓℓℓ),(7.37)
and we perform the change of variable t→ τ in (7.36), and we assume for a moment that
(7.38)
Here R(t, xℓℓℓ) = Õ(e−µk+1t) is expandible. As in [18, Section 5] and [5, Section 6], we get
e−t ∼
− 2µk
(z−k )
(z+ℓ )
eµks(xℓℓℓ)
)−1/µkτ1/µk
τ bµj/µkbj(− ln τ, xℓℓℓ)
(7.39)
t ∼− 1
ln τ +
− 2µk
(z−k )
(z+ℓ )
eµks(xℓℓℓ)
τ bµj/µkbj(− ln τ, xℓℓℓ)(7.40)
τ bµj/µkbj(− ln τ, xℓℓℓ),(7.41)
where the bj ’s change from line to line. These expansions are valid in the following sense:
Definition 7.1. Let f(τ, y) be defined on ]0, ε[×U where U ⊂ Rm. We say that f = Ô(g(τ))
(resp. f = ô(g(τ))), where g(τ) is a non-negative function defined in ]0, ε[ if and only if for
all α ∈ N and β ∈ Nm,
(7.42) (τ∂τ )
α∂βy f(τ, y) = O(g(τ)),
(resp. o(g(τ))) for all (τ, y) ∈]0, ε[×U .
Thus, an expression like f ∼
j=1 τ
bµj/µkfj(− ln τ, xℓℓℓ), where fj(− ln τ, xℓℓℓ) is a polynomial
with respect to ln τ , like in (7.39)–(7.41), means that, for all J ∈ N,
(7.43) f(τ, x)−
τ bµj/µkfj(− ln τ, xℓℓℓ) = Ô(τ bµJ/µk).
We shall say that such symbols f are called expandible near 0.
Since f ℓ0(t, xℓℓℓ, h) is expandible (see Definition 6.1) with respect to t, this symbol is also
expandible near 0 with respect to τ in the previous sense. In particular, we get
(7.44) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ
τ (Σ(E)+cµj)/µk f̃ ℓ0,j(− ln τ, xℓℓℓ),
where the f̃ ℓ0,j’s are polynomials with respect to ln τ . The principal symbol f̃
0,0 is independent
on ln τ and we have
(7.45) f̃ ℓ0,0(xℓℓℓ) =
− 2µk
eµks(xℓℓℓ)
)−Σ(E)/µkf ℓ0,0(xℓℓℓ).
In that case, (7.36) becomes
(7.46) Asing0 =
c(E)h−1/2
(2π)1−n/2
∫∫ +∞
eiτ/hf̃ ℓ0(τ, xℓℓℓ)
dxℓℓℓ +O(h∞).
50 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1,
we can perform the integration with respect to t of each term in the right hand side of (7.44),
modulo a term O(h∞) (see (D.3)–(D.4) in Lemma D.1). Then, we get
(7.47) Asing0 =
c(E)h−1/2
(2π)1−n/2
f̂j(ln h)h
(Σ(E)+bµj )/µk ,
where f̂j(lnh) is a polynomial in respect to lnh. f̂0 does not depend on h and we have
(7.48) f̂0 = Γ(Σ(E)/µk)(−i)−Σ(E)/µk
f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ.
To finish the proof, it remains to perform the integration with respect to xℓℓℓ in (7.48). From
(7.22) and (7.45), it becomes
f̂0 =Γ(Σ(E)/µk)
(z−k )
(z+ℓ )
eµks(xℓℓℓ)
)−Σ(E)/µk
× α0,0(yℓ
xℓℓℓ)
ã+,0
yℓ(xℓℓℓ)
π/2 e
i sgnΦ′′
|Hℓxℓℓℓ
(yℓ(xℓℓℓ))π/4
∣∣ detΦ′′|Hℓxℓℓℓ
yℓ(xℓℓℓ)
)∣∣1/2
dxℓℓℓ.(7.49)
Now we make the change of variable xℓℓℓ 7→ s given by yℓ(xℓℓℓ) = x+ℓ (s) (then s(xℓℓℓ) = s). In
particular,
(7.50) dxℓℓℓ = ∂s(x
ℓ,ℓℓℓ(s))ds = λℓℓℓ|g
ℓℓℓ (z
ℓ )|e
λℓℓℓs(1 + o(1))ds,
as s→ −∞. In this setting, we get
(7.51) α0,0(x
(s)) = α0,0(0)(1 + o(1)),
as s→ −∞, where α0,0(0) is given in (6.8). We also have, from (7.9) and (7.13),
(7.52) ã+,0(x
ℓ (s)) = −i∂s
χ̃+(γ
ℓ (s))
(2E0)
1/4(D+ℓ )
−1/2eisz.
Then, putting (7.23), (7.24), (7.50), (7.51) and (7.52) in (7.49), we obtain
f̂0 =Γ(Σ(E)/µk)
(z−k )
(z+ℓ )
〉)−Σ(E)/µkα0,0(0)∂s
χ̃+(γ
ℓ (s))
i(n−1)π/4
j 6=ℓℓℓ
λℓℓℓ|g+ℓℓℓ (z
)|(2E0)1/4(D+ℓ )
−1/2eisze−Σ(E)seλℓℓℓs(1 + o(1)) ds
i(n+1)π/4
j 6=ℓℓℓ
)−1/2
λℓℓℓ|g+ℓℓℓ (z
)|Γ(Σ(E)/µk)
〉)−Σ(E)/µk
× e−iν
π/2α0,0(0)(2E0)
1/4(D+ℓ )
χ̃+(γ
ℓ (s))
(1 + o(1)) ds.(7.53)
Here the o(1) does not depend on χ̃+. Now, we choose a family of cut-off functions (χ̃
+)j∈N
such that the support of ∂t
goes to −∞ as j → +∞. We also assume that
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 51
ℓ (t))
is non negative (see Figure 1). Then
f̂0 =−
ei(n+1)π/4
j 6=ℓℓℓ
)−1/2
λℓℓℓΓ(Σ(E)/µk)e
π/2eiπ/4(2λ1)
3/2e−iν
× |g−1 (z
)| |g+
〉)−Σ(E)/µk(7.54)
(2E0)
1/2(D−
)−1/2 × (1 + o(1)).(7.55)
as j → +∞. Since f̂0 is also independent of χ̃+, we obtain Theorem 2.6 from (7.47) and
(7.48), in the case (a) and under the assumption (7.38). When 〈g−
(z−k )|g
(z+ℓ )〉 > 0, we set
τ as the opposite of the R.H.S. of (7.37), and we obtain the result along the same lines (see
Remark D.2).
Now we assume that we are in the case (b) of the assumption (A7). In that case, the
phase function Ψ is given by (7.30). For xℓℓℓ fixed in a compact set outside from 0, we set,
mimicking (7.37),
yℓ(t, xℓℓℓ)
− (S−k + S
=M2(k, ℓ)e2λ1s(xℓℓℓ)t2e−2λ1t +R(t, xℓℓℓ)(7.56)
where R(t, xℓℓℓ) = O(te−2λ1t) is expandible with respect to t. As above, we assume that
M2(k, ℓ) is positive (the other case can be studied the same way).
Following (7.39), we want to write s := e−t as a function of τ . Since t 7→ τ(t) is expandible
with respect to t, we have
(7.57) τ = M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ)),
where r(s, xℓℓℓ) = ô(1). In particular, ∂sτ > 0 for s positive small enough and then, for ε > 0
small enough, s 7→ τ(s) is invertible for 0 < s < ε. We denote s(τ) the inverse of this function.
We look for s(τ) of the form
(7.58) s(τ) = (2λ1)
M2(k, ℓ)e2λ1s(xℓℓℓ)
)1/2λ1 u(τ, xℓℓℓ)
(− ln τ)1/λ1
where u(τ, xℓℓℓ) has to be determined. Using (7.57), the equation on u is
τ =M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ))
=τu2λ1
(2λ1)
−2M2(k, ℓ)e2λ1s(xℓℓℓ)
+ 2λ1
− 2ln(− ln τ)
1 + r
(2λ1)
M2(k, ℓ)e2λ1s(xℓℓℓ)
)1/2λ1 u
(− ln τ)1/λ1
, xℓℓℓ
=τF (τ, u, xℓℓℓ),(7.59)
where F = u2λ1(1 + r̃(τ, u, xℓℓℓ)) and r̃ = ô(1) for u close to 1 (here (u, xℓℓℓ) are the variables y
in Definition 7.1). In other word, to find u, we have to solve F (t, u, xℓℓℓ) = 1.
First we remark that u 7→ F (τ, u, xℓℓℓ) is real-valued and continuous. Since, for δ > 0 and
τ small enough, F (τ, 1 − δ, xℓℓℓ) < 1 < (τ, 1 + δ, xℓℓℓ), there exists u ∈ [1 − δ, 1 + δ] such that
F (τ, 1 + δ, xℓℓℓ) = 1. Thank to the discussion before (7.58), the function s(τ) is of the form
(7.58) with u(τ, xℓℓℓ) ∈ [1− δ, 1 + δ], for τ small enough.
52 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
For τ > 0, the function F is C∞ and, since r̃ = ô(1), we have
(7.60) ∂u
F (τ, u, xℓℓℓ)− 1
(u(τ, xℓℓℓ)) = 2λ1u
2λ1−1(1 + oτ (1)) > λ1,
for τ small enough. The notation oτ (1) means a term which goes to 0 as τ goes to 0. Here
we have used the fact that u(τ, xℓℓℓ) is close to 1. In particular, the implicit function theorem
implies that u(τ, xℓℓℓ) is C
We write u = 1 + v(τ, xℓℓℓ) and we known that v ∈ C∞ and v = oτ (1). Differentiating the
equality
(7.61) 1 = F (τ, u(τ, xℓℓℓ), xℓℓℓ) =
u(τ, xℓℓℓ)
)2λ1(
1 + r̃(τ, u(τ, xℓℓℓ), xℓℓℓ)
one can show that v = ô(1). Thus we have
e−t =s(τ) = (2λ1)
M2(k, ℓ)e2λ1s(xℓℓℓ)
)1/2λ1 1 + r̂(τ, xℓℓℓ)
(− ln τ)1/λ1
,(7.62)
t =− ln τ
(1 + r̂(τ, xℓℓℓ)),(7.63)
+ r̂(τ, xℓℓℓ),(7.64)
where r̂(τ, xℓℓℓ) = ô(1) change from line to line.
Since f ℓ0(t, xℓℓℓ, h) is expandible with respect to t, we get, from (7.62)–(7.64),
(7.65) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ
= τΣ(E)/2λ1(− ln τ)−Σ(E)/λ1
f̃ ℓ0,0(xℓℓℓ) + r̂(τ, xℓℓℓ)
where r̂ = ô(1) and
(7.66) f̃ ℓ0,0(xℓℓℓ) = (2λ1)
Σ(E)/λ1−1
M2(k, ℓ)e2λ1s(xℓℓℓ)
)−Σ(E)/2λ1
f ℓ0,0(xℓℓℓ).
In that case, (7.36) becomes
(7.67) Asing0 =
c(E)h−1/2
(2π)1−n/2
∫∫ +∞
eiτ/hf̃ ℓ0(τ, xℓℓℓ)
dxℓℓℓ +O(h∞).
Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1,
we can perform the integration with respect to t in (7.67), modulo an error term given by
(D.3)–(D.4) in Lemma D.1. Then, we get
Asing0 =
c(E)h−1/2
(2π)1−n/2
)/hΓ(Σ(E)/2λ1)(−i)−Σ(E)/2λ1
× hΣ(E)/2λ1(− lnh)−Σ(E)/λ1
f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ + o(1)
,(7.68)
as h goes to 0. The rest of the proof follows that of (7.55).
At last, the proof of Theorem 2.6 in the case (c) can be obtained along the same lines, and
we omit it.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 53
Appendix A. Proof of Proposition 2.5
We prove that Λ+θ ∩Λ+ 6= ∅. From the assumption (A2), the Lagrangian manifold Λ+ can
be described, near (0, 0) ∈ T ∗(Rn), as
(A.1) Λ+ = {(x, ξ); x = ∇ϕ̃+(ξ)},
for |ξ| < 2ε, with ε > 0 small enough. For η ∈ Sn−1, let (x(t, η), ξ(t, η)) be the bicharacteristic
curve with initial condition (ϕ̃(εη), εη). We have
(A.2) Λ+ = {(x(t, η), ξ(t, η)); t ∈ R, η ∈ Sn−1} ∪ {(0, 0)}.
The function ξ(t, η) is continuous on R× Sn−1. From the classical scattering theory (see [13,
Section 1.3]), we know that this function ξ(t, η) converges uniformly to
(A.3) ξ(∞, η) := lim
ξ(t, η),
as t→ +∞ and ξ(∞, η) ∈
2E Sn−1.
Then, the function
(A.4) F (t, η) =
1−t , η)
|ξ( t
1−t , η)|
is well defined for 0 ≤ t ≤ 1 with the convention F (1, η) = ξ(∞, η)/
2E. Here we used that
|ξ(t, η)| 6= 0 for each t ∈ [0,+∞], η ∈ Sn−1. The previous properties of ξ(t, η) imply the
continuity of F (t, η) on [0, 1] × Sn−1.
From (A.2), to prove that Λ+θ ∩ Λ+ 6= ∅ for all θ ∈ Sn−1, it is enough (equivalent) to show
the surjectivity of η → F (1, η). But if η → F (1, η) is not onto, then ImF (1, ·) ⊂ Sn−1 \ {a
point}. And since Sn−1 \ {a point} is a contractible space, F (1, ·) is homotopic to a constant
(A.5) f : Sn−1 → Sn−1.
On the other hand, F : [0, 1]× Sn−1 −→ Sn−1 gives a homotopy between F (0, ·) = IdSn−1 and
F (1, ·). In particular, we have
(A.6) 1 = deg(F (0, ·)) = deg(F (1, ·)) = deg(f(·)) = 0,
which is impossible (see [16, Section 23] for more details).
Appendix B. A lower bound for the resolvent
Let χ ∈ C∞(]0,+∞[) be a non-decreasing function such that
(B.1) χ(x) =
x for 0 < x < 1
2 for 2 < x,
Let also ϕ ∈ C∞0 (R) an even function such that 0 ≤ ϕ ≤ 1, 1[−1,1] ≺ ϕ, and suppϕ ⊂ [−2, 2].
We set
(B.2) u(x) =
j/2hϕ
|xj |1/2
uj(x),
54 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where 0 < α < 2β will be fixed later on. The uj ’s are of course C
∞ functions, and we have
(B.3) (P − E0)u = −
∆u(x)−
x2ju(x) +O(x3u(x)).
Lemma B.1. For any h small enough, we have
hβn| lnh|n/2 . ‖u‖L2(Rn) . hβn| lnh|n/2,(B.4)
∥∥|x|3u(x)
L2(Rn)
. h3αhβn| lnh|n/2.(B.5)
Proof. First of all, the second estimate follow easily from the first one: we have
∥∥|x|3u(x)
∥∥2 =
|x|6|u(x)|2dx . h6α‖u‖2,
since u vanishes if |x| > 2hα. Thanks to the fact that u is a product of n functions of one
variable, it is enough to estimate
dt = 2
∫ 2hα
We have
dt ≤ I ≤ 2
∫ 2hα
dt+ 2
so that
dt ≤ I ≤ 2
∫ 2hα
dt+ 2
4 dt.
The first estimate follows from the fact that 2β − α > 0, once we have noticed that
∫ Ahα
dt = h2β
(2β − α)| ln h|+ α lnA
On the other hand, we have
∆u(x)−
x2ju(x) =
j 6=k
uj(xj)
u′′k(xk)−
x2kuk(xk)
From Lemma B.1, we get
∥∥(P − E0)u
∥∥ .hβ(n−1)| lnh|(n−1)/2 sup
1≤k≤n
∥∥h2u′′k(t) + λ
2uk(t)
∥∥+ h3αhβn| lnh|n/2
h−β | lnh|−1/2 sup
1≤k≤n
∥∥h2u′′k(t) + λ
2uk(t)
∥∥+ h3α
‖u‖.(B.6)
We also have
(B.7) h2u′′k(t) + λkt
2uk(t) = e
h2v′′h(t) + ihλk(2t∂t + 1)vh(t)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 55
where we have set vh(t) = ϕ
|t|1/2
. Notice that the right hand side of (B.7) is an even
function, so that we only have to consider t > 0. The point here, is that we have, for t > 0,
(B.8) (2t∂t + 1)
= − h


2 if 0 < t <
O(1) if h
< t < h2β ,
0 if h2β < t.
Therefore, we obtain
∥∥(2t∂t + 1)vh
∥∥2 =2
∫ 2hα
(2t∂t + 1)
|t|1/2
∫ 2hα
|t|1/2
∫ h2β
∫ 2hα
|t|1/2
dt . h2β .(B.9)
On the other hand, an easy computation gives, still for t > 0,
v′′h(t) =h
−2αϕ′′
4t5/2
.(B.10)
Computing the L2–norm of each of these terms as in Lemma B.1 and (B.9), we obtain
(B.11) ‖h2v′′h‖ . h2+β−2α + h2+β−2α + h2−3β + h2−3β ,
and, eventually, from (B.6), (B.7), (B.9) and (B.11),
∥∥(P − E0)u
h−β | lnh|−1/2
h1+β + h2+β−2α + h2−3β
+ h3α
Therefore we obtain Proposition 2.2 if we can find α > 0 and β > 0 such that
2− 2α > 1, 2− 4β > 1, 3α > 1 and 2β > α,
and one can check that α = 5/12 and β = 11/48 satisfies these four inequalities.
Appendix C. Lagrangian manifolds which are transverse to Λ±
Let Λ ⊂ p−1(E0) be a Lagrangian manifold such that Λ ∩ Λ− is transverse along a Hamil-
tonian curve γ(t) = (x(t), ξ(t)). Then, where exists a 6= 0 and ν ∈ {1, . . . , n} such that
(C.1) γ(t) = (a+O(e−εt))e−λν t,
as t→ +∞. The vector a is an eigenvector of
(C.2)
V ′′(0) 0
for the eigenvalue λν . Thus, up to a linear change of variable in R
n, we can always assume
that Πxa is collinear to the xν–direction. The goal of this section is to prove the following
geometric result.
56 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Proposition C.1. For t large enough, Λ projects nicely on Rnx near γ(t). In particular,
there exists ψ ∈ C∞(Rn) defined near Πxγ, unique up to a constant, such that Λ = Λψ :=
{(x,∇ψ(x)); x ∈ Rn}. Moreover, we have
(C.3) ψ′′(x(t)) =

. . .
. . .

+O(e−εt),
as t→ +∞.
Remark C.2. The same result hold in the outgoing region: If γ = Λ ∩ Λ+ is transverse,
Λ projects nicely on Rnx near γ(t), t → −∞. Then Λ = Λψ for some function ψ satisfying
ψ′′(x(t)) = diag(−λ1, . . . ,−λν−1, λν ,−λν+1, . . . ,−λn) +O(eεt).
Proof. We follow the proof of [18, Lemma 2.1]. There exist symplectic local coordinates (y, η)
centered at (0, 0) such that Λ− (resp. Λ+) is given by y = 0 (resp. η = 0) and
(ξj + λjxj) +O((x, ξ)2),(C.4)
(ξj − λjxj) +O((x, ξ)2).(C.5)
Then, p(x, ξ) = A(y, η)y · η with A0 := A(0, 0) = diag(λ1, . . . , λn).
(C.6)
A0 +O(e−λ1t) 0
O(e−λ1t) A0 +O(e−λ1t)
We denote by U(t, s) the linear operator such that U(t, s)δ solves (C.6) with U(s, s) = Id.
Since Λ∩Λ− = γ is transverse, there exists En−1(t0) ⊂ Tγ(t0)Λ, a vector space of dimension
n − 1 disjoint from Tγ(t0)Λ−. For convenience, we set En(t0) = En−1(t0) ⊕ Rv for some
v /∈ Tγ(t0)Λ + Tγ(t0)Λ−. Let E•(t) = U(t, t0)E•(t0). From [18, Lemma 2.1], there exists Bt =
O(e−λ1t) such that En(t) is given by δη = Btδx. Now, if δ ∈ En−1(t), we have σ(Hp, δ) = 0
since En−1(t)⊕ RHp = Tγ(t)Λ and Λ is a Lagrangian manifold. From (C.1), we have
(C.7) Hp(γ(t)) = γ̇(t) = −λν(ãeην +O(e−εt))e−λν t,
where eην is the basis vector corresponding to ην and then
(C.8) 0 = σ(eλν tHp, δ) = λν ãδyν +O(e−εt)|δ|.
It follows that δ ∈ En−1(t) if and only if (δyν , δη) = B̃tδy′ with B̃t = O(e−εt). Using Tγ(t)Λ =
En−1(t)⊕ RHp, we obtain that Tγ(t)Λ has a basis formed of vector fj(t) such that
fj =eyj +O(e−εt) for j 6= ν(C.9)
fν =eην +O(e−εt).(C.10)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 57
In the (x, ξ)-coordinates, Tγ(t)Λ has a basis formed of vector f̃j(t) of the form
f̃j =eξj + λjexj +O(e−εt) for j 6= ν(C.11)
f̃ν =eξν − λjexν +O(e−εt),(C.12)
and the lemma follows. �
Appendix D. Asymptotic behaviour of certain integrals
Lemma D.1. Let α ∈ C, Reα > 0, β ∈ R and χ ∈ C∞0 (]−∞, 1/2[) be such that χ = 1 near
0. As λ goes to +∞, we have
(D.1)
eiλttα(− ln t)βχ(t) dt
= Γ(α)(ln λ)β(−iλ)−α(1 + o(1)).
Moreover, if β ∈ N, we get
(D.2)
eiλttα(− ln t)βχ(t) dt
= (−iλ)−α
(j)(α)(−1)j
ln(−iλ)
+O(λ−∞).
Finally, if s(t) ∈ C∞(]0,+∞[) satisfies
(D.3) |∂jt s(t)| = o
tα−j(− ln t)β
for all j ∈ N and t→ 0, then
(D.4)
eiλts(t)χ(t)
(ln λ)βλ−α
Here (−iλ)−α = eiαπ/2λ−α and ln(−iλ) = lnλ− iπ/2.
Remark D.2. Notice that one obtains the behaviour of these quantities as λ → −∞ by
taking the complex conjugate in these expressions.
Proof. We begin with (D.2) and assume first that β = 0. Then, we can write
eiλttαχ(t)
= lim
ei(λ+iε)ttαχ(t)
= lim
I1(α, ε) + I2(α, ε)
,(D.5)
where
I1(α, ε) =
e−(ε−iλ)ttα
,(D.6)
I2(α, ε) =
ei(λ+iε)ttα(1− χ(t)) dt
·(D.7)
It is clear that
(D.8) I1(α, ε) = (ε− iλ)−αΓ(α),
where z−α is well defined on C\]−∞, 0] and real positive on ]0,+∞[. In particular
(D.9) lim
I1(α, ε) = (−iλ)−αΓ(α).
58 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Concerning, I2(α, ε), we remark that r(t, α) = t
α−1(1− χ(t)) is a symbol which satisfies
(D.10) |∂jt ∂kαr(t, α)| . 〈t〉Reα−1−j〈ln t〉k,
for all j, k ∈ N uniformly for t ∈ [0,+∞[ and α in a compact set of {Re z > 0}. Then, making
integration by parts in (D.7), we obtain
(D.11) I2(α, ε) =
(ε− iλ)j
e(iλ−ε)t∂jt r(t, α) dt,
for all j ∈ N. Now, if j is large enough (j > Reα), ∂jt r(t, α) is integrable in time uniformly
with respect to ε. In particular, for such j,
(D.12) lim
I2(α, ε) = e
ijπ/2λ−j
eiλt∂
t r(t, α) dt,
and then (see (D.10) or Cauchy’s formula)
∂kα lim
I2(α, ε) =e
ijπ/2λ−j
eiλt∂
αr(t, α) dt
=O(λ−∞),(D.13)
for all k ∈ N. Then we obtain (D.2) for β = 0. To obtain the result for β ∈ N, it is enough
to see that
eiλttα(ln t)βχ(t)
eiλttαχ(t)
(−iλ)−αΓ(α)
+ ∂βα lim
I2(α, ε)
=(−iλ)−α
(j)(α)
− ln(−iλ)
+O(λ−∞),(D.14)
from (D.13). Thus, (D.2) is proved.
Let u be a function C∞(]0,+∞[) be such that
(D.15) |∂jt u(t)| . tReα−j(− ln t)β,
near 0. Let ϕ ∈ C∞(R) such that ϕ = 1 for t < 1 and ϕ = 0 for t > 2. For δ > 0, we have
(D.16)
eiλtu(t)χ(t)
1−ϕ(t/δ)
= (−iλ)−N
eiλt∂Nt
u(t)χ(t)
1−ϕ(t/δ)
for all N .
If one of the derivatives falls on 1−ϕ(t/δ), the support of this contribution is inside [δ, 2δ].
Therefore, the corresponding term will be bounded by δReα−N−1(ln δ)β and will contribute
like δReα−N (− ln δ)β to the integral.
If ones of the derivatives falls on χ(t), the support of the integrand will be a compact set
outside of 0 and then this function will be O(1). The contribution to the integral of such
term will be like 1.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 59
If all the derivatives fall on u(t)t−1, we corresponding term will satisfies
eiλt∂Nt
u(t)t−1
1− ϕ(t/δ)
dt =O(1)
tReα−1−N (− ln t)β(1− χ(t))dt
.(− ln δ)βδReα−N ,(D.17)
for N large enough (N > Reα).
From this 3 cases, we deduce
(D.18)
eiλtu(t)χ(t)
1− ϕ(t/δ)
(− ln δ)βδα−Nλ−N
Taking δ = (ελ)−1, we get
(D.19)
eiλtu(t)χ(t)
1− ϕ(t/δ)
ε(lnλ)βλ−α
as λ→ +∞.
We now assume (D.3), and we want to prove (D.4). Since, for t small enough
(D.20) tReα−1(− ln t)β .
tReα(− ln t)β
we get
eiλts(t)χ(t)ϕ(t/δ)
∣∣∣ =oδ→0(1)
tReα−1(− ln t)βdt
=oδ→0(1)δ
Re α(− ln δ)β .(D.21)
Here oδ→0(1) stands for a term which goes to 0 as δ goes to 0. If δ = (ελ)
−1, we get
(D.22)
eiλts(t)χ(t)ϕ(t/δ)
∣∣∣ = oλ→+∞(1)λ−α(ln λ)β,
when λ → +∞ and ε fixed. Taking ε small enough in (D.19), and then λ large enough in
(D.22), we get (D.4).
We are left with (D.1). We need to compute
(D.23) I =
eiλttα(− ln t)βϕ(t/δ) dt
Performing the change of variable s = λt, we get
I =λ−α
∫ 2/ε
eissα(ln λ− ln s)βϕ(εs) ds
=(lnλ)βλ−α
∫ 2/ε
eissα(1− ln s/ lnλ)βϕ(εs) ds
.(D.24)
60 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
We remark that, in the previous equation, − ln s/ lnλ > − ln(2/ε)/ ln λ > −1/2 for λ large
enough. Using (1 + u)β = 1 +O(|u|+ |u|max(1,β)) for u > −1/2, we get
I =(lnλ)βλ−α
∫ 2/ε
eissαϕ(εs)
+ (lnλ)βλ−α
∫ 2/ε
sReαO
( | ln s|
( | ln s|
)max(1,β))
ϕ(εs)
=(lnλ)β
eiλttα(− ln t)βϕ(t/δ) dt
(lnλ)β−1λ−α
.(D.25)
Note that the Oε in (D.25) depends on ε.
Then, using (D.19), (D.25) and (D.19) again, we get
eiλttα(− ln t)βχ(t) dt
=I +O
ε(ln λ)βλ−α
=(ln λ)β
eiλttα(− ln t)βϕ(t/δ) dt
(lnλ)β−1λ−α
ε(lnλ)βλ−α
=(ln λ)β
eiλttα(− ln t)β dt
ε(lnλ)βλ−α
(lnλ)β−1λ−α
ε(ln λ)βλ−α
.(D.26)
Choosing ε small enough, then λ large enough, and using (D.2) with β = 0 to compute the
first term, we obtain
(D.27)
eiλttα(− ln t)βχ(t) dt
= Γ(α)(ln λ)β(−iλ)−α(1 + o(1)),
and this finishes the proof for (D.1). �
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http://arxiv.org/abs/math/0602069
62 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Ivana Alexandrova, Department of Mathematics, East Carolina University, Greenville, NC
27858, USA
E-mail address: alexandrovai@ecu.edu
Jean-François Bony, Institut de Mathématiques de Bordeaux, (UMR CNRS 5251), Université
de Bordeaux I, 33405 Talence, France
E-mail address: bony@math.u-bordeaux1.fr
Thierry Ramond, Mathématiques, Université Paris Sud, (UMR CNRS 8628), 91405 Orsay,
France
E-mail address: thierry.ramond@math.u-psud.fr
	1. Introduction
	2. Assumptions and main results
	3. Proof of the main resolvent estimate
	4. Representation of the Scattering Amplitude
	5. Computations before the critical point
	5.1.  Computation of u- in the incoming region
	5.2.  Computation of u- along -k
	6. Computation of u- at the critical point
	6.1.  Study of the transport equations for the phases
	6.2.  Taylor expansions of + and k1
	6.3.  Asymptotics near the critical point for the trajectories
	6.4.  Computation of the jk's
	7. Computations after the critical point
	7.1.  Stationary phase expansion in the outgoing region
	7.2.  Behaviour of the phase function 
	7.3.  Integration with respect to time
	Appendix A. Proof of Proposition 2.5
	Appendix B. A lower bound for the resolvent
	Appendix C. Lagrangian manifolds which are transverse to 
	Appendix D. Asymptotic behaviour of certain integrals
	References

  We compute the scattering amplitude for Schr\"odinger operators at a critical
energy level, corresponding to the maximum point of the potential. We follow
the wrok of Robert and Tamura, '89, using Isozaki and Kitada's representation
formula for the scattering amplitude, together with results from Bony, Fujiie,
Ramond and Zerzeri '06 in order to analyze the contribution of trapped
trajectories.

Introduction 2
2. Assumptions and main results 4
3. Proof of the main resolvent estimate 10
4. Representation of the Scattering Amplitude 16
5. Computations before the critical point 19
5.1. Computation of u− in the incoming region 19
5.2. Computation of u− along γ
6. Computation of u− at the critical point 23
6.1. Study of the transport equations for the phases 25
6.2. Taylor expansions of ϕ+ and ϕ
6.3. Asymptotics near the critical point for the trajectories 34
6.4. Computation of the ϕkj ’s 36
7. Computations after the critical point 43
7.1. Stationary phase expansion in the outgoing region 43
7.2. Behaviour of the phase function Φ 47
Date: April 12, 2007.
2000 Mathematics Subject Classification. 81U20,35P25,35B38,35C20.
Key words and phrases. Scattering amplitude, critical energy, Schrödinger equation.
Acknowledgments: We would like to thank Johannes Sjöstrand for helpful discussions during the prepa-
ration of this paper. The first author also thanks Victor Ivrii for supporting visits to Université Paris Sud,
Orsay, and the Department of Mathematics at Orsay for the extended hospitality.
http://arxiv.org/abs/0704.1632v2
2 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
7.3. Integration with respect to time 48
Appendix A. Proof of Proposition 2.5 53
Appendix B. A lower bound for the resolvent 53
Appendix C. Lagrangian manifolds which are transverse to Λ± 55
Appendix D. Asymptotic behaviour of certain integrals 57
References 60
1. Introduction
We study the semiclassical behavior of scattering amplitude at energy E > 0 for Schrö-
dinger operators
(1.1) P (x, hD) = −h
∆ + V (x)
where V is a real valued C∞ function on Rn, which vanishes at infinity. We shall suppose
here that E is close to a critical energy level E0 for P , which corresponds to a non-degenerate
global maximum of the potential. Here, we address the case where this maximum is unique.
Let us recall that, if V (x) = O(〈x〉−ρ) for some ρ > (n + 1)/2, then for any ω 6= θ ∈ Sn−1
and E > 0, the problem
P (x, hD)u = Eu,
u(x, h) = ei
2Ex·ω/h +A(ω, θ,E, h)e
2E|x|/h
|x|(n−1)/2
+ o(|x|(1−n)/2) as x→ +∞, x|x| = θ,
has a unique solution. The scattering amplitude at energy E for the incoming direction ω
and the outgoing direction θ is the real number A(ω, θ,E, h).
For potentials that are not decaying that fast at infinity, it is not that easy to write down a
stationary formula for the scattering amplitude: If V (x) = O(〈x〉−ρ) for some ρ > 1, one can
define the scattering matrix at energy E using wave operators (see Section 4 below). Then,
writing
(1.2) S(E, h) = Id− 2iπT (E, h),
one can see that T (E, h) is a compact operator on L2(Sn−1), which kernel T (ω, θ,E, h) is
smooth out of the diagonal in Sn−1×Sn−1. Then, the scattering amplitude is given for θ 6= ω,
(1.3) A(ω, θ,E, h) = c(E))h(n−1)/2T (ω, θ,E, h),
where
(1.4) c(E) = −2π(2E)−
4 (2π)
2 e−i
(n−3)π
We proceed here as in [30], where D. Robert and H. Tamura have studied the semiclassical
behavior of the scattering amplitude for short range potentials at a non-trapping energy E .
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 3
An energy E is said to be non-trapping when K(E), the trapped set K(E) at energy E, is
empty. This trapped set is defined as
(1.5) K(E) =
(x, ξ) ∈ p−1(E), exp(tHp)(x, ξ) 6→ ∞ as t→ ±∞
where Hp is the Hamiltonian vector field associated to the principal symbol p(x, ξ) =
V (x) of the operator P . Notice that the scattering amplitude has been first studied, in the
semiclassical regime, by B. Vainberg [32] and Y. Protas [27] in the case of compactly supported
potential, and for non-trapping energies, where they obtained the same type of result.
Under the non-trapping assumption, and some other non-degeneracy condition (in fact our
assumption (A4) below), D. Robert and H. Tamura have shown that the scattering amplitude
has an asymptotic expansion with respect to h. The non-degeneracy assumption implies in
particular that there is a finite number N∞ of classical trajectories for the Hamiltonian p,
with asymptotic direction ω for t→ −∞ and asymptotic direction θ as t→ +∞. Robert and
Tamura’s result is the following asymptotic expansion for the scattering amplitude:
(1.6) A(ω, θ,E, h) =
iS∞j /h
aj,m(ω, θ,E)h
m +O(h∞), h→ 0,
where S∞j is the classical action along the corresponding trajectory. Also, they have computed
the first term in this expansion, showing that it can be given in terms of quantities attached
to the corresponding classical trajectory only.
There are also some few works concerning the scattering amplitude when the non-trapping
assumption is not fulfilled. In his paper [24], L. Michel has shown that, if there is no trapped
trajectory with incoming direction ω and outgoing direction θ (see the discussion after (2.6)
below), and if there is a complex neighborhood of E of size ∼ hN for some N ∈ N possibly
large, which is free of resonances, then A(ω, θ,E, h) is still given by Robert and Tamura’s
formula. The potential is also supposed to be analytic in a sector out of a compact set, and
the assumption on the existence of a resonance free domain around E amounts to an estimate
on boundary value of the meromorphic extension of the truncated resolvent of the for
(1.7) ‖χ(P − (E ± i0))−1χ‖ = O(h−N ), χ ∈ C∞0 (Rn).
Of course, these assumptions allow the existence of a non-empty trapped set.
In [2] and [3], the first author has shown that at non-trapping energies or in L. Michel’s
setting, the scattering amplitude is an h-Fourier Integral Operator associated to a natural
scattering relation. These results imply that the scattering amplitude admits an asymptotic
expansion even without the non-degeneracy assumption, and in the sense of oscillatory inte-
grals. In particular, the expansion (1.6) is recovered under the non-degeneracy assumption
and as an oscillatory integral.
In [21], A. Lahmar-Benbernou and A. Martinez have computed the scattering amplitude
at energy E ∼ E0, in the case where the trapped set K(E0) consists in one single point
corresponding to a local minimum of the potential (a well in the island situation). In that
case, the estimate (1.7) is not true, and their result is obtained through a construction of the
resonant states.
In the present work, we compute the scattering amplitude at energy E ∼ E0 in the case
where the trapped set K(E0) corresponds to the unique global maximum of the potential.
4 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
The one-dimensional case has been studied in [28, 14, 15], with specific techniques, and we
consider here the general n > 1 dimensional case.
Notice that J. Sjöstrand in [31], and P. Briet, J.-M. Combes and P. Duclos in [7, 8] have
described the resonances close to E0 in the case where V is analytic in a sector around R
From their result, it follows that Michel’s assumption on the existence of a not too small
resonance-free neighborhood of E0 is satisfied. However, we show below (see Proposition 2.5)
that for any ω ∈ Sn−1, there is at least one half-trapped trajectory with incoming direction
ω, so that L. Michel’s result never applies here.
Here, we do not assume analyticity for V . We compute the contributions to the scattering
amplitude arising from the classical trajectories reaching the unstable equilibrium point, which
corresponds to the top of the potential barrier. At the quantum level, tunnel effect occurs,
which permits the particle to pass through this point. Our computation here relies heavily
on [5], where a precise description of this phenomena has been obtained. In a forthcoming
paper, we shall show that in this case also, the scattering amplitude is an h-Fourier Integral
Operator.
This paper is organized in the following way. In Section 2, we describe our assumptions, and
state our main results: a resolvent estimate, and the asymptotic expansion of the scattering
amplitude in the semiclassical regime. Section 3 is devoted to the proof of the resolvent
estimate, from which we deduce in Section 4 estimates similar to those in [30]. In that
section, we also recall briefly the representation formula for the scattering amplitude proved
by Isozaki and Kitada, and introduce notations from [30]. The computation of the asymptotic
expansion of the scattering amplitude is conducted in sections 5, 6 and 7, following the classical
trajectories. Eventually, we have put in four appendices the proofs of some side results or
technicalities.
2. Assumptions and main results
We suppose that the potential V satisfies the following assumptions
(A1) V is a C∞ function on Rn, and, for some ρ > 1,
∂αV (x) = O(〈x〉−ρ−|α|).
(A2) V has a non-degenerate maximum point at x = 0, with E0 = V (0) > 0 and
∇2V (0) =
. . .
 , 0 < λ1 ≤ λ2 ≤ . . . ≤ λn.
(A3) The trapped set at energy E0 is K(E0) = {(0, 0)}.
Notice that the assumptions (A1)–(A3) imply that V has an absolute global maximum
at x = 0. Indeed, if L = {x 6= 0; V (x) ≥ E0} was non empty, the geodesic, for the
Agmon distance (E0 − V (x))1/2+ dx, between 0 and L would be the projection of a trapped
bicharacteristic (see [1, Theorem 3.7.7]).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 5
As in D. Robert and H. Tamura in [30], one of the key ingredient for the study of the
scattering amplitude is a suitable estimate for the resolvent. Using the ideas in [5, Section 4],
we have obtained the following result, that we think to be of independent interest.
Theorem 2.1. Suppose assumptions (A1), (A2) and (A3) hold, and let α > 1
be a fixed
real number. We have
(2.1) ‖P − (E ± i0))−1‖α,−α . h−1| lnh|,
uniformly for |E − E0| ≤ δ, with δ > 0 small enough. Here ‖Q‖α,β denotes the norm of the
bounded operator Q from L2(〈x〉α dx) to L2(〈x〉β dx).
Moreover, we prove in the Appendix B that our estimate is not far from optimal. Indeed,
we have the
Proposition 2.2. Under the assumptions (A1) and (A2), we have
(2.2) ‖(P − E0 ± i0)−1‖α,−α & h−1
| lnh|.
We would like to mention that in the case of a closed hyperbolic orbit, the same upper
bound has been obtained by N. Burq [9] in the analytic category, and in a recent paper [11]
by H. Christianson in the C∞ setting.
As a matter of fact, in the present setting, S. Nakamura has proved in [26] an O(h−2)
bound for the resolvent. Nakamura’s estimate would be sufficient for our proof of Theorem
2.6, but it is not sharp enough for the computation of the total scattering cross section along
the lines of D. Robert and H. Tamura in [29]. In that paper, the proof relies on a bound
O(h−1) for the resolvent, but it is easy to see that an estimate like O(h−1−ε) for any small
enough ε > 0 is sufficient. If we denote
(2.3) σ(ω,E0, h) =
|A(ω, θ,E, h)|2dθ,
the total scattering cross-section, and following D. Robert and H. Tamura’s work, our resolvent
estimates gives the
Theorem 2.3. Suppose assumptions (A1), (A2) and (A3) hold, and that ρ > n+1
, n ≥ 2.
If |E − E0| < δ for some δ > 0 small enough, then
(2.4) σ(ω,E, h) = 4
2−1(2E)−1/2h−1
V (y + sω)ds
dy +O(h−(n−1)/(ρ−1)).
Now we state our assumptions concerning the classical trajectories associated with the
Hamiltonian p, that is curves t 7→ γ(t, x, ξ) = exp(tHp)(x, ξ) for some initial data (x, ξ) ∈
T ∗Rn. Let us recall that, thanks to the decay of V at infinity, for given α ∈ Sn−1 and
z ∈ α⊥ ∼ Rn−1 (the impact plane), there is a unique bicharacteristic curve
(2.5) γ±(t, z, α,E) = (x±(t, z, α,E), ξ±(t, z, α,E))
such that
(2.6)
|x±(t, z, α,E) −
2Eαt− z| = 0,
|ξ±(t, z, α,E) −
2Eα| = 0.
6 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
We shall denote by Λ−ω the set of points in T
n lying on trajectories going to infinity with
direction ω as t → −∞, and Λ+θ the set of those which lie on trajectories going to infinity
with direction θ as t→ +∞:
(2.7)
Λ−ω =
γ−(t, z, ω,E) ∈ T ∗Rn, z ∈ ω⊥, t ∈ R
Λ+θ =
γ+(t, z, θ, E) ∈ T ∗Rn, z ∈ θ⊥, t ∈ R
We shall see that Λ−ω and Λ
θ are in fact Lagrangian submanifolds of T
Under the assumptions (A1), (A2) and (A3) there are only two possible behaviors for
x±(t, z, α,E) as t→ ∓∞: either it escapes to ∞, or it goes to 0.
First we state our assumptions for the first kind of trajectories. For these, we also have,
for some ξ∞(z, ω,E),
ξ−(t, z, ω) = ξ∞(z, ω,E),
and we shall say that the trajectory γ−(t, z, ω,E) has initial direction ω and final direction
θ = ξ∞(z, ω,E)/2
E. As in [30] we shall suppose that there is only a finite number of
trajectories with initial direction ω and final direction θ. This assumption can be given in
terms of the angular density
(2.8) σ̂(z) = |det(ξ∞(z, ω,E), ∂z1ξ∞(z, ω,E), . . . , ∂zn−1ξ∞(z, ω,E))|.
Definition 2.4. The outgoing direction θ ∈ Sn−1 is called regular for the incoming direction
ω ∈ Sn−1, or ω-regular, if θ 6= ω and, for all z′ ∈ ω⊥ with ξ∞(z′, ω,E) = 2
Eθ, the map
ω⊥ ∋ z 7→ ξ∞(z, ω,E) ∈ Sn−1 is non-degenerate at z′, i.e. σ̂(z′) 6= 0.
We fix the incoming direction ω ∈ Sn−1, and we assume that
(A4) the direction θ ∈ Sn−1 is ω-regular.
Then, one can show that Λ−ω ∩ Λ+θ is a finite set of Hamiltonian trajectories (γ∞j )1≤j≤N∞ ,
γ∞j (t) = γ
∞(t, z∞j ) = (x
j (t), ξ
j (t)), with transverse intersection along each of these curves.
We turn to trapped trajectories. Let us notice that the linearization Fp at (0, 0) of the
Hamilton vector field Hp has eigenvalues −λn, . . . ,−λ1, λ1, . . . , λn. Thus (0, 0) is a hyper-
bolic fixed point for Hp, and the stable/unstable manifold Theorem gives the existence of
a stable incoming Lagrangian manifold Λ− and a stable outgoing Lagrangian manifold Λ+
characterized by
(2.9) Λ± = {(x, ξ) ∈ T ∗Rn, exp(tHp)(x, ξ) → 0 as t→ ∓∞} .
In this paper, we shall describe the contribution to the scattering amplitude of the trapped
trajectories, that is those going from infinity to the fixed point (0, 0). We have proved in
Appendix A the following result, which shows that there are always such trajectories.
Proposition 2.5. For every ω, θ ∈ Sn−1, we have
(2.10) Λ−ω ∩ Λ− 6= ∅ and Λ+θ ∩ Λ+ 6= ∅.
We suppose that
(A5) Λ−ω and Λ− (resp. Λ
and Λ+) intersect transversally.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 7
Under this assumption, Λ−ω ∩ Λ− and Λ+θ ∩ Λ+ are finite sets of bicharacteristic curves. We
denote them, respectively,
(2.11) γ−k : t 7→ γ
−(t, z−k ) = (x
k (t), ξ
−(t)), 1 ≤ k ≤ N−,
(2.12) γ+ℓ : t 7→ γ
+(t, z+ℓ ) = (x
+(t), ξ+(t)), 1 ≤ ℓ ≤ N+.
Here, the z−
(resp. the z+
) belong to ω⊥ (resp. θ⊥) and determine the corresponding curve
by (2.6).
We recall from [18, Section 3] (see also [5, Section 5]), that each integral curve γ±(t) =
(x±(t), ξ±(t)) ∈ Λ± satisfies, in the sense of expandible functions (see Definition 6.1 below),
(2.13) γ±(t) ∼
γ±j (t)e
±µj t, as t→ ∓∞,
where µ1 = λ1 < µ2 < . . . is the strictly increasing sequence of linear combinations over N of
the λj’s. Here, the functions γ
j : R → R2n are polynomials, that we write
(2.14) γ±j (t) =
M ′j∑
γ±j,mt
Considering the base space projection of these trajectories, we denote
(2.15) x±(t) ∼
g±j (t)e
±µj t, as t→ ∓∞, g±j (t) =
M ′j∑
g±j,mt
Let us denote ̂ the (only) integer such that µb = 2λ1. We prove in Proposition 6.11 below that
if j < ̂, then M ′j = 0, or more precisely, that g
j (t) = g
j is a constant vector in Ker(Fp∓λj).
We also have M ′
≤ 1, and g−
can be computed in terms of g−1 .
In this paper, concerning the incoming trajectories, we shall assume that,
(A6) For each k ∈ {1, . . . , N−}, g−1 (z
) 6= 0.
Finally, we state our assumptions for the outgoing trajectories γ+ℓ ⊂ Λ+ ∩Λθ+. First of all,
it is easy to see, using Hartman’s linearization theorem, that there exists always a m ∈ N
such that g+m(z
ℓ ) 6= 0. We denote
(2.16) ℓℓℓ = ℓℓℓ(ℓ) = min{m, g+m(z+ℓ ) 6= 0}
the smallest of these m’s. We know that µℓℓℓ is one of the λj’s, and that M
ℓℓℓ = 0.
In [5], we have been able to describe the branching process between an incoming curve
γ− ⊂ Λ− and an outgoing curve γ+ ⊂ Λ+ provided 〈g−1 |g
1 〉 6= 0 (see the definition for
Λ̃+(ρ−) before [5, Theorem 2.6]). Here, for the computation of the scattering amplitude, we
can relax a lot this assumption, and analyze the branching in other cases that we describe
now. Let us denote, for a given pair of paths (γ−(z−k ), γ
+(z+ℓ )) in (Λ
ω ∩ Λ−)× (Λ+θ ∩ Λ+),
(2.17) M2(k, ℓ) = −
j∈I1(2λ1)
α,β∈I2(λ1)
βV (0)
(g−1 (z
αV (0)
(g+1 (z
8 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
M1(k, ℓ) =−
α∈I2(λ1)
αV (0)
(g−1 (z
−))α(g+
(z+))j + (g
(z−))j(g
α,β∈I2(λ1)
(g−1 (z
(g+1 (z
Cα,β,(2.18)
where
Cα,β =− ∂α+βV (0) +
j∈I1\I1(2λ1)
λ2j(4λ
1 − λ2j)
∂α+γV (0)∂β+γV (0)
γ,δ∈I2(λ1)
γ+δ=α+β
(γ + δ)!
γ! δ!
γV (0)∂j∂
δV (0).(2.19)
Here, we have set I1 = {1, . . . , n}, 1j = (δij)i=1,...,n ∈ Nn and
(2.20) Im(µ) = {β ∈ Nn, β = 1k1 + · · ·+ 1km with λk1 = · · · = λkm = µ},
the set of multi-indices β of length |β| = m with each index of its non-vanishing components
in the set {j ∈ N, λj = µ}. We also denote Im ⊂ Nn the set of multi-indices of length m.
We will suppose that
(A7) For each pair of paths (γ−(z−k ), γ
+(z+ℓ )), k ∈ {1, . . . , N−}, ℓ ∈ {1, . . . , N+}, one of the
three following cases occurs:
(a) The set
m < ̂, 〈g−m(z−k )|g+m(z
ℓ )〉 6= 0
is not empty. Then we denote
k = min
m < ̂, 〈g−m(z−k )|g
)〉 6= 0
(b) For all m < ̂, we have 〈g−m(z−k )|g+m(z
ℓ )〉 = 0, and M2(k, ℓ) 6= 0.
(c) For all m < ̂, we have 〈g−m(z−k )|g+m(z
ℓ )〉 = 0, M2(k, ℓ) = 0 and M1(k, ℓ) 6= 0.
As one could expect (see [30], [28] or [15]), action integrals appear in our formula for the
scattering amplitude. We shall denote
S∞j =
(|ξ∞j (t)|2 − 2E0)dt, j ∈ {1, . . . , N∞},(2.21)
S−k =
|ξ−k (t)|
2 − 2E01t<0
dt, k ∈ {1, . . . , N−},(2.22)
(t)|2 − 2E01t>0
dt, ℓ ∈ {1, . . . , N+},(2.23)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 9
and ν∞j , ν
ℓ , ν
k the Maslov indexes of the curves γ
j , γ
ℓ , γ
k respectively. Let also
D−k = limt→+∞
∣∣∣ det
∂x−(t, z, ω,E0)
∂(t, z)
∣∣∣ e−(Σλj−2λ1)t,(2.24)
D+ℓ = limt→−∞
∣∣∣ det
∂x+(t, z, ω,E0)
∂(t, z)
∣∣∣ e(Σλj−2λℓℓℓ)t,(2.25)
be the Maslov determinants for γ−
, and γ+
respectively. We show below that 0 < D−
+∞. Eventually we set
(2.26) Σ(E, h) =
− iE − E0
Then, the main result of this paper is the
Theorem 2.6. Suppose assumptions (A1) to (A7) hold, and that E ∈ R is such that
E −E0 = O(h). Then
A(ω, θ,E, h) =
Aregj (ω, θ,E, h) +
Asingk,ℓ (ω, θ,E, h) +O(h
∞),(2.27)
where
(2.28) Aregj (ω, θ,E, h) = e
iS∞j /h
j,m(ω, θ,E)h
j,0 (ω, θ,E) =
−iν∞j π/2
σ̂(zj)1/2
Moreover we have
• In case (a)
Asingk,ℓ (ω, θ,E, h) = e
k,ℓ,m(ω, θ,E, lnh)h
(Σ(E)+bµm)/µk−1/2,(2.29)
where the a
k,ℓ,m
(ω, θ,E, ln h) are polynomials with respect to lnh, and
k,ℓ,0(ω, θ,E, ln h) =
π1−n/2
ei(nπ/4−π/2)
)−1/2
(Σ(E)
(2λ1λℓℓℓ)
× e−iν
π/2e−iν
π/2(D−
)−1/2
× |g−1 (z
k )| |g
ℓℓℓ (z
(z−k )
(z+ℓ )
〉)−Σ(E)/µk .(2.30)
• In case (b)
(2.31) Asingk,ℓ (ω, θ,E, h) = e
k,ℓ (ω, θ,E)
hΣ(E)/2λ1−1/2
| lnh|Σ(E)/λ1
(1 + o(1)),
10 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where
(ω, θ,E) =
π1−n/2
ei(nπ/4−π/2)
)−1/2
(Σ(E)
(2λ1λℓℓℓ)
3/2(2λ1)
Σ(E)/λ1−1
× e−iν
π/2e−iν
π/2(D−k D
× |g−1 (z
)| |g+
− iM2(k, ℓ)
)−Σ(E)/2λ1
.(2.32)
• In case (c)
(2.33) Asingk,ℓ (ω, θ,E, h) = e
k,ℓ (ω, θ,E)
hΣ(E)/2λ1−1/2
| ln h|Σ(E)/2λ1
(1 + o(1)),
where
k,ℓ (ω, θ,E) =
π1−n/2
ei(nπ/4−π/2)
)−1/2
(Σ(E)
(2λ1λℓℓℓ)
3/2(2λ1)
Σ(E)/2λ1−1
× e−iν
π/2e−iν
π/2(D−k D
× |g−1 (z
k )| |g
ℓℓℓ (z
− iM1(k, ℓ)
)−Σ(E)/2λ1
.(2.34)
Here, the µ̂j are the linear combinations over N of the λk’s and λk − λ1’s, and the function
z 7→ z−Σ(E)/µk is defined on C\]−∞, 0] and real positive on ]0,+∞[.
Of course the assumption that 〈g−1 |g
1 〉 6= 0 (a subcase of (a)) is generic. Without the
assumption (A4), the regular part Areg of the scattering amplitude has an integral rep-
resentation as in [3]. When the assumption (A7) is not fulfilled, that is when the terms
corresponding to the µj with j ≤ ̂ do not contribute, we don’t know if the scattering
amplitude can be given only in terms of the g±1 ’s and of the derivatives of the potential.
3. Proof of the main resolvent estimate
Here we prove Theorem 2.1 using Mourre’s Theory. We start with the construction of an
escape function close to the stationary point (0, 0) in the spirit of [10] and [5]. Since Λ+ and
Λ− are Lagrangian manifolds, one can choose local symplectic coordinates (y, η) such that
(3.1) p(x, ξ) = B(y, η)y · η,
where (y, η) 7→ B(y, η) is a C∞ mapping from a neighborhood of (0,0) in T ∗Rn to the space
Mn(R) of n× n matrices with real entries, such that,
(3.2) B(0, 0) =
. . .
We denote U a unitary Fourier Integral Operator (FIO) microlocally defined in a neighborhood
of (0, 0), which canonical transformation is the map (x, ξ) 7→ (y, η), and we set
(3.3) P̂ = UPU∗.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 11
Here the FIO U∗ is the adjoint of U , and we have UU∗ = Id+O(h∞) and U∗U = Id+O(h∞)
microlocally near (0, 0). Then P̂ is a pseudodifferential operator, with a real (modulo O(h∞))
symbol p̂(y, η) =
j p̂j(y, η)h
j , such that
(3.4) p̂0 = B(y, η)y · η.
We set B1 = Oph(b1),
(3.5) b1(y, η) =
χ̃2(y, η),
where M > 1 will be fixed later and χ̃1 ≺ χ̃2 ∈ C∞0 (T ∗Rn) with χ̃1 = 1 near (0, 0). In what
follows, we will assume that hM < 1. In particular, b1 ∈ S1/2(| ln h|). Here and in what
follows, we use the usual notation for classes of symbols. For m an order function, a function
a(x, ξ, h) ∈ C∞(T ∗Rn) belongs to Sδh(m) when
(3.6) ∀α ∈ N2n, ∃Cα > 0, ∀h ∈]0, 1], |∂αx,ξa(x, ξ, h)| ≤ Cαh−δ|α|m(x, ξ).
Let us also recall that, if a ∈ Sα(1) and b ∈ Sβ(1), with α, β < 1/2, we have
(3.7)
Oph(a),Oph(b)
= Oph
ih{b, a}
+ h3(1−α−β) Oph(r),
with r ∈ Smin(α,β)(1): In particular the term of order 2 vanishes.
Hence, we have here
(3.8) [B1, P̂ ] = Oph
ih{p̂0, b1}
+ | lnh|h3/2 Oph(rM ),
with rM ∈ S1/2(1). The semi-norms of rM may depend on M . We have
(3.9) {p̂0, b1} = c1 + c2,
{p̂0, χ̃2}(3.10)
p̂0, ln
By + (∂ηB)y · η
hM + y2
Bη + (∂yB)y · η
hM + η2
χ̃2.(3.11)
The symbols c1 ∈ S1/2(| lnh|), c2 ∈ S1/2(1) satisfy supp(c1) ⊂ supp(∇χ̃2). Let ϕ̃ ∈
C∞0 (T
n) be a function such that ϕ̃ = 0 near (0, 0) and ϕ̃ = 1 near the support of ∇χ̃2. We
Oph(c1) =Oph(ϕ̃)Oph(c1)Oph(ϕ̃) +O(h∞)
≥− C1h| ln h|Oph(ϕ̃)Oph(ϕ̃) +O(h∞)
≥− C1h| ln h|Oph(ϕ̃2) +O(h2| ln h|),(3.12)
for some C1 > 0. On the other hand, using [5, (4.96)–(4.97)], we get
(3.13) Oph(c2) ≥ εM−1 Oph(χ̃1) +O(M−2),
12 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
for some ε > 0. With the notation A1 = U
∗B1U , the formulas (3.8), (3.9), (3.12) and (3.13)
imply
−i[A1, P ] =− iU∗[B1, P ]U
≥εhM−1U∗Oph(χ̃1)U − C1h| lnh|U∗ Oph(ϕ̃2)U
+O(hM−2) +OM (h3/2| lnh|).(3.14)
If κ is the canonical transformation associated to U , then χj = χ̃j ◦ κ, j = 1, 2 and ϕ = ϕ̃ ◦ κ
are C∞0 (T
∗(Rn), [0, 1]) functions which satisfy χ1 = 1 near (0, 0) and ϕ = 0 near (0, 0). Using
Egorov’s Theorem, (3.14) becomes
(3.15) − i[A1, P ] ≥ εhM−1 Oph(χ1)− C1h| lnh|Oph(ϕ) +O(hM−2) +OM (h3/2| lnh|).
Now, we build an escape function outside of supp(χ1) as in [22]. Let 1(0,0) ≺ χ0 ≺
χ1 ≺ χ2 ≺ χ3 ≺ χ4 ≺ χ5 be C∞0 (T ∗(Rn), [0, 1]) functions with ϕ ≺ χ4. We define a3 =
g(ξ)(1−χ3(x, ξ))x ·ξ where g ∈ C∞0 (Rn) satisfies 1p−1([E0−δ,E0+δ]) ≺ g. Using [6, Lemma 3.1],
we can find a bounded, C∞ function a2(x, ξ) such that
(3.16) Hpa2 ≥
0 for all (x, ξ) ∈ p−1([E0 − δ,E0 + δ]),
1 for all (x, ξ) ∈ supp(χ4 − χ0) ∩ p−1([E0 − δ,E0 + δ]),
and we set A2 = Oph(a2χ5). We denote
(3.17) A = A1 + C2| lnh|A2 + | lnh|A3,
where C2 > 1 will be fixed later. Now let ψ̃ ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]) with ψ̃ = 1 near
E0. We recall that ψ̃(P ) is a classical pseudodifferential operator of class Ψ
0(〈ξ〉−∞) with
principal symbol ψ̃(p). Then, from (3.15), we obtain
−iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1ψ̃(P )Oph(χ1)ψ̃(P )− C1h| ln h|ψ̃(P )Oph(ϕ)ψ̃(P )
+ C2h| lnh|Oph
ψ̃2(p)(χ4 − χ0)
+ C2h| ln h|Oph
ψ̃2(p)a2Hpχ5
+ h| ln h|Oph
ψ̃2(p)(ξ2 − x · ∇V )(1 − χ3)
+ h| ln h|Oph
ψ̃2(p)x · ξHp(gχ3)
+O(hM−2) +OM (h3/2| lnh|).(3.18)
From (A1), we have x·∇V (x) → 0 as x→ ∞. In particular, if χ3 is equal to 1 in a sufficiently
large zone, we have
(3.19) ψ̃2(p)(ξ2 − x · ∇V )(1− χ3) ≥ E0ψ̃2(p)(1− χ3).
If C2 > 0 is large enough, the G̊arding inequality implies
(3.20)
C2 Oph
ψ̃2(p)(χ4 − χ0)
−C1 Oph
ψ̃2(p)ϕ
ψ̃2(p)x · ξHp(gχ3)
≥ Oph
ψ̃2(p)(χ4 − χ0)
+O(h).
As in [22], we take χ5(x) = χ̃5(µx) with µ small and χ̃5 ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]). Since
a2 is bounded, we get
(3.21)
∣∣C2ψ̃2(p)a2Hpχ5
∣∣ ≤ µC2‖a2‖L∞‖Hpχ̃5‖L∞ . µ.
Therefore, if µ is small enough, (3.19) implies
(3.22) Oph
ψ̃2(p)(ξ2−x ·∇V )(1−χ3)
+C2 Oph
ψ̃2(p)a2Hpχ5
ψ̃2(p)(1−χ3)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 13
Then (3.18), (3.20), (3.22) and the G̊arding inequality give
−iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1 Oph
ψ̃2(p)χ1
+ h| ln h|Oph
ψ̃2(p)(χ4 − χ0)
h| ln h|Oph
ψ̃2(p)(1 − χ3)
+O(hM−2) +OM (h3/2| lnh|)
≥εhM−1 Oph
ψ̃2(p)
+O(hM−2) +OM (h3/2| ln h|).(3.23)
Choosing M large enough and 1E0 ≺ ψ ≺ ψ̃, we have proved the
Lemma 3.1. Let M be large enough and ψ ∈ C∞0 ([E0− δ,E0+ δ]), δ > 0 small enough, with
ψ = 1 near E0. Then, we have
(3.24) − iψ(P )[A,P ]ψ(P ) ≥ εh−1ψ2(P ).
Moreover
(3.25) [A,P ] = O(h| ln h|).
From the properties of the support of the χj, we have
[[P,A], A] =[[P,A1], A1] + C2| lnh|[[P,A1], A2]
+ C2| ln h|[[P,A2], A1] + C22 | lnh|2[[P,A2], A2] + C2| lnh|2[[P,A2], A3]
+ C2| ln h|2[[P,A3], A2] + | lnh|2[[P,A3], A3] +O(h∞).(3.26)
We also know that P ∈ Ψ0(〈ξ〉2), A2 ∈ Ψ0(〈ξ〉−∞) and A3 ∈ Ψ0(〈x〉〈ξ〉−∞). Then, we can
show that all the terms in (3.26) with j, k = 2, 3 satisfy
(3.27) [[P,Aj ], Ak] ∈ Ψ0(h2).
On the other hand,
(3.28) [[P,A1], A2] = U
∗[[P̂ , B1], UA2U
∗]U +O(h∞),
with UA2U
∗ ∈ Ψ0(1). From (3.8) – (3.11), we have [P̂ , B1] ∈ Ψ1/2(h| ln h|) and then
(3.29) [[P,A1], A2] = O(h3/2| lnh|).
The term [[P,A2], A1] gives the same type of contribution. It remains to study
(3.30) [[P,A1], A1] = U
∗[[P̂ , B1], B1]U +O(h∞).
Let χ̃3 ∈ C∞0 (T ∗Rn), [0, 1]) with χ̃2 ≺ χ̃3 and
(3.31) f =
χ̃3(y, η) ∈ S1/2(| ln h|).
Then, with a remainder rM ∈ S1/2(1) which differs from line to line,
i[P̂ , B1] =hOph
f{χ̃2, p̂0}+ c2
− h3/2| ln h|Oph(rM )
=hOph(f)Oph({χ̃2, p̂0}) + hOph(c2) + h3/2| lnh|Oph(rM ).(3.32)
14 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
In particular, since [P̂ , B1] ∈ Ψ1/2(h| ln h|), c2 ∈ S1/2(1) and f ∈ S1/2(| lnh|),
[[P̂ , B1], B1] =[[P̂ , B1],Oph(fχ̃2)]
=− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(fχ̃2)]− ih[Oph(c2),Oph(fχ̃2)]
+O(h3/2| ln h|2)
=− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(f)Oph(χ̃2)] +O(h| ln h|)
=− ihOph(f)[Oph({χ̃2, p̂0}),Oph(f)]Oph(χ̃2)
− ih[Oph(f),Oph(f)]Oph({χ̃2, p̂0})Oph(χ̃2)
− ihOph(f)Oph(f)[Oph({χ̃2, p̂0}),Oph(χ̃2)]
− ihOph(f)[Oph(f),Oph(χ̃2)]Oph({χ̃2, p̂0}) +O(h| ln h|)
=O(h| ln h|).(3.33)
From (3.26), (3.27), (3.29) and (3.33), we get
(3.34) [[P,A], A] = O(h| lnh|).
As a matter of fact, using [5], one can show that [[P,A], A] = O(h). Now we can use the
following proposition which is an adaptation of the limiting absorption principle of Mourre
[25] (see also [12, Theorem 4.9], [19, Proposition 2.1] and [4, Theorem 7.4.1]).
Proposition 3.2. Let (P,D(P )) and (A,D(A)) be self-adjoint operators on a separable
Hilbert space H. Assume the following assumptions:
i) P is of class C2(A). Recall that P is of class Cr(A) if there exists z ∈ C \ σ(P ) such
(3.35) R ∋ t→ eitA(P − z)−1e−itA,
is Cr for the strong topology of L(H).
ii) The form [P,A] defined on D(A) ∩D(P ) extends to a bounded operator on H and
(3.36) ‖[P,A]‖ . β.
iii) The form [[P,A],A] defined on D(A) extends to a bounded operator on H and
(3.37) ‖[[P,A],A]‖ . γ.
iv) There exist a compact interval I ⊂ R and g ∈ C∞0 (R) with 1I ≺ g such that
(3.38) ig(P )[P,A]g(P ) & γg2(P ).
v) β2 . γ . 1.
Then, for all α > 1/2, limε→0〈A〉−α(P − E ± iε)−1〈A〉−α exists and
(3.39)
∥∥〈A〉−α(P − E ± i0)−1〈A〉−α
∥∥ . γ−1,
uniformly for E ∈ I.
Remark 3.3. From Theorem 6.2.10 of [4], we have the following useful characterization of
the regularity C2(A). Assume that (ii) and (iv) hold. Then, P is of class C2(A) if and only
if, for some z ∈ C \ σ(P ), the set {u ∈ D(A); (P − z)−1u ∈ D(A) and (P − z)−1u ∈ D(A)}
is a core for A.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 15
Proof. The proof follows the work of Hislop and Nakamura [19]. For ε > 0, we define M2 =
ig(P )[P,A]g(P ) and Gε(z) = (P − iεM2 − z)−1 which is analytic for Re z ∈ I and Im z > 0.
Following [12, Lemma 4.14] with (3.35)), we get
(3.40) ‖g(P )Gε(z)ϕ‖ . (εγ)−1/2|(ϕ,Gε(z)ϕ)|1/2,
(3.41) ‖(1− g(P ))Gε(z)‖ . 1 + εβ‖Gε(z)‖,
and then
(3.42) ‖Gε(z)‖ . (εγ)−1,
for ε < ε0 with ε0 small enough, but independent on β, γ.
As in [19], let Dε = (1 + |A|)−α(1 + ε|A|)α−1 for α ∈]1/2, 1] and Fε(z) = DεGε(z)Dε. Of
course, from (3.42),
(3.43) ‖Fε(z)‖ . (εγ)−1,
and (3.40) and (3.41) with ϕ = Dεψ give
(3.44) ‖Gε(z)Dε‖ . 1 + (εγ)−1/2‖Fε‖1/2.
The derivative of Fε(z) is given by (see [12, Lemma 4.15])
(3.45) ∂εFε(z) = iDεGεM
2GεDε = Q0 +Q1 +Q2 +Q3,
Q0 =(α− 1)|A|(1 + |A|)−α(1 + ε|A|)α−2Gε(z)Dε
+ (α− 1)DεGε(z)|A|(1 + |A|)−α(1 + ε|A|)α−2(3.46)
Q1 =DεGε(1− g(P ))[P,A](1 − g(P ))GεDε(3.47)
Q2 =DεGε(1− g(P ))[P,A]g(P )GεDε +DεGεg(P )[P,A](1 − g(P ))GεDε(3.48)
Q3 =−DεGε[P,A]GεDε.(3.49)
From (3.44), we obtain
(3.50) ‖Q0‖ . εα−1
1 + (εγ)−1/2‖Fε‖1/2
and from (3.36), v) of Proposition 3.2, (3.41) and (3.42), we get
(3.51) ‖Q1‖ . γ−1.
Using in addition (3.44), we obtain
(3.52) ‖Q2‖ . 1 + (εγ)−1/2‖Fε‖1/2.
Now we write Q3 = Q4 +Q5 with
Q4 = −DεGε[P − iεM2 − z,A]GεDε(3.53)
Q5 = −iεDεGε[M2,A]GεDε.(3.54)
For Q4, we have the estimate
(3.55) ‖Q4‖ . εα−1
1 + (εγ)−1/2‖Fε‖1/2
On the other hand, (3.36), (3.37) and v) imply
(3.56) ‖[M2,A]‖ . γ.
16 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Then (3.44) gives
(3.57) ‖Q5‖ . 1 + ‖Fε‖.
Using the estimates on the Qj, we get
(3.58) ‖∂εFε‖ . εα−1
γ−1 + (εγ)−1/2‖Fε‖1/2 + ‖Fε‖
Using (3.43) and integrating (3.37) N times with respect to ε, we get
(3.59) ‖Fε‖ . γ−1
1 + ε2α(1−2
−N )−1),
so that, for N large enough,
(3.60) lim sup
‖〈A〉−α(P − E ± iδ)−1〈A〉−α‖ . γ−1.
Using, as in [19], that z 7→ F0(z) is Hölder continuous, we prove the existence of the limit
limIm z→0 F0(z) for Re z ∈ I and the proposition follows from (3.60). �
From Lemma 3.1 and (3.34), we can apply Proposition 3.2 with A = A/| ln h|, β = h and
γ = h/| ln h|. Therefore we have the estimate
(3.61)
∥∥〈A〉−α(P − E ± i0)−1〈A〉−α
∥∥ . h−1| ln h|,
for E ∈ [E0 − δ,E0 + δ]. As usual, we have
(3.62) ‖〈x〉−α〈A〉α‖ = O(1),
for α ≥ 0. Indeed, (3.62) is clear for α ∈ 2N, and the general case follows by complex
interpolation. Then, (3.61) and (3.26) imply Theorem 2.1.
4. Representation of the Scattering Amplitude
As in [30], our starting point for the computation of the scattering amplitude is the rep-
resentation given by Isozaki and Kitada in [20]. We recall briefly their formula, that they
obtained writing parametrices for the wave operators W± as Fourier Integral Operators, tak-
ing advantage of the well-known intertwining property W±P = P0W±, P = P0 + V . The
wave operators are defined by
(4.1) W± = s− lim
eitP/he−itP0/h,
where the limit exist thanks to the short-range assumption (A1). The scattering operator
is by definition S = (W+)∗W−, and the scattering matrix S(E, h) is then given by the
decompostion of S with respect to the spectral measure of P0 = −h2∆. Now we recall briefly
the discussion in [30, Section 1,2] (see also [3]), and we start with some notations.
If Ω is an open subset of T ∗Rn , we denote by Am(Ω) the class of symbols a such that
(x, ξ) 7→ a(x, ξ, h) belongs to C∞(Ω) and
(4.2)
∣∣∣∂αx ∂
ξ a(x, ξ)
∣∣∣ ≤ Cαβ〈x〉m−|α|〈ξ〉−L, for all L > 0, (x, ξ) ∈ Ω, (α, β) ∈ Nd × Nd.
We also denote by
(4.3) Γ±(R, d, σ) =
(x, ξ) ∈ Rn × Rn : |x| > R, 1
< |ξ| < d,± cos(x, ξ) > ±σ
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 17
with R > 1, d > 1, σ ∈ (−1, 1), and cos(x, ξ) = 〈x,ξ〉|x| |ξ| , the outgoing and incoming subsets
of T ∗Rn, respectively. Eventually, for α > 1
, we denote the bounded operator F0(E, h) :
L2α(R
n) → L2(Sn−1) given by
(4.4) (F0(E, h)f) (ω) = (2πh)−
2 (2E)
2E〈ω,x〉f(x)dx,E > 0.
Isozaki and Kitada have constructed phase functions Φ± and symbols a± and b± such that,
for some R0 >> 0, 1 < d4 < d3 < d2 < d1 < d0, and 0 < σ4 < σ3 < σ2 < σ1 < σ0 < 1:
i) Φ± ∈ C∞(T ∗Rn) solve the eikonal equation
(4.5)
|∇xΦ±(x, ξ)|2 + V (x) =
in (x, ξ) ∈ Γ±(R0, d0,±σ0), respectively.
ii) (x, ξ) 7→ Φ±(x, ξ) − x · ξ ∈ A0 (Γ±(R0, d0,±σ0)) .
iii) For all (x, ξ) ∈ T ∗Rn
(4.6)
∂xj∂ξk
(x, ξ) − δjk
∣∣∣∣ < ε(R0),
where δjk is the Kronecker delta and ε(R0) → 0 as R0 → +∞.
iv) a± ∼
j=0 h
ja±j , where a±j ∈ A−j(Γ±(3R0, d1,∓σ1)), supp a±j ⊂ Γ±(3R0, d1,∓σ1),
a±j solve
(4.7) 〈∇xΦ±|∇xa±0〉+
(∆xΦ±) a±0 = 0
(4.8) 〈∇xΦ±|∇xa±j〉+
(∆xΦ±) a±j =
∆xa±j−1, j ≥ 1,
with the conditions at infinity
(4.9) a±0 → 1, a±j → 0, j ≥ 1, as |x| → ∞.
in Γ±(2R0, d2,∓σ2), and solve (4.7) and (4.8) in Γ±(4R0, d1,∓σ2).
v) b± ∼
j=0 h
jb±j, where b±j ∈ A−j(Γ±(5R0, d3,±σ4), supp b±j ⊂ Γ±(5R0, d3,±σ4),
b±j solve (4.7) and (4.8) with the conditions at infinity (4.9) in Γ±(6R0, d4,±σ3), and
solve (4.7) and (4.8) in Γ±(6R0, d3,±σ3).
For a symbol c and a phase function ϕ, we denote by Ih(c, ϕ) the oscillatory integral
(4.10) Ih(c, ϕ) =
(2πh)n
(ϕ(x,ξ)−〈y,ξ〉)c(x, ξ)dξ
and we set
(4.11)
K±a(h) = P (h)Ih(a±,Φ±)− Ih(a±,Φ±)P0(h),
K±b(h) = P (h)Ih(b±,Φ±)− Ih(b±,Φ±)P0(h).
The operator T (E, h) for E ∈] 1
[ is then given by (see [20, Theorem 3.3])
(4.12) T (E, h) = T+1(E, h) + T−1(E, h) − T2(E, h),
18 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where
(4.13) T±1(E, h) = F0(E, h)Ih(a±,Φ±)∗K±b(h)F∗0 (E, h)
(4.14) T2(E, h) = F0(E, h)K∗+a(h)R(E + i0, h) (K+b(h) +K−b(h))F∗0 (E, h),
where we denote from now on R(E ± i0, h) = (P − (E ± i0))−1.
Writing explicitly their kernel, it is easy to see, by a non-stationary phase argument, that
the operators T±1 are O(h∞) when θ 6= ω. Therefore we have
(4.15) A(ω, θ,E, h) = −c(E)h(n−1)/2T2(ω, θ,E, h) +O(h∞),
where c(E) is given in (1.4).
As in [30], we shall use our resolvent estimate (Theorem 2.1) in a particular form. It was
noticed by L. Michel in [24, Proposition 3.1] that, in the present trapping case, the following
proposition follows easily from the corresponding one in the non-trapping setting. Indeed, if
ϕ is a compactly supported smooth function, it is clear that P̃ = −h2∆+ (1− ϕ(x/R))V (x)
satisfies the non-trapping assumption for R large enough, thanks to the decay of V at ∞.
Writing [30, Lemma 2.3] for P̃ , one gets the
Proposition 4.1. Let ω± ∈ A0 has support in Γ±(R, d, σ±) for R > R0. For E ∈ [E0 −
δ,E0 + δ], we have
(i) For any α > 1/2 and M > 1, then, for any ε > 0,
(4.16) ‖R(E ± i0, h)ω±(x, hDx)‖−α+M,−α = O(h−3−ε).
(ii) If σ+ > σ−, then for any α≫ 1,
(4.17) ‖ω∓(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞).
(iii) If ω(x, ξ) ∈ A0 has support in |x| < (9/10)R, then for any α≫ 1
(4.18) ‖ω(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞).
Then we can follow line by line the discussion after Lemma 2.1 of D. Robert and H. Tamura,
and we obtain (see Equations 2.2-2.4 there):
(4.19) A(ω, θ,E, h) = c(E)h−(n+1)/2〈R(E + i0, h)g−eiψ−/h, g+eiψ+/h〉+O(h∞),
where
(4.20) g± = e
−iψ±/h[χ±, P ]a±(x, h)e
iψ±/h,
(4.21) ψ+(x) = Φ+(x,
2Eθ), ψ−(x) = Φ−(x,
2Eω).
Moreover the functions χ± are C
n) functions such that χ± = 1 on some ball B(0, R±),
with support in B(0, R± + 1).
Eventually, we shall need the following version of Egorov’s Theorem, which is also used in
Robert and Tamura’s paper.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 19
Proposition 4.2 ([30, Proposition 3.1]). Let ω(x, ξ) ∈ A0 be of compact support. Assume
that, for some fixed t ∈ R, ωt is a function in A0 which vanishes in a small neighborhood of
{(x, ξ); (x, ξ) = exp(tHp)(y, η), (y, η) ∈ suppω}.
‖Oph(ωt)e−itP/hOph(ω)‖−α,α = O(h∞),
for any α ≫ 1. Moreover, the order relation is uniform in t when t ranges over a compact
interval of R.
In the three next sections, we prove Theorem 2.6 using (4.19). We set
(4.22) u− = u
− = R(E + i0, h)g−eiψ−/h,
and our proof consists in the computation of u− in different region of the phase space, following
the classical trajectories γ∞j , or γ
k and γ
ℓ . It is important to notice that we have (P−E)u− =
0 out of the support of g−.
5. Computations before the critical point
5.1. Computation of u− in the incoming region.
We start with the computation of u− in an incoming region which contains the micro-
support of g−. Notice that, thanks to Theorem 2.1, 〈x〉−αu−(x) is a semiclassical family of
distributions for α > 1/2.
Lemma 5.1. Let P be a Schrödinger operator as in (1.1) satisfying only (A1). Suppose that
I is a compact interval of ]0,+∞[, and d > 0 is such that I ⊂] 1
[. Suppose also that
0 < σ+ < 1, R is large enough and K ⊂ T ∗Rn is a compact subset of {|x| > R} ∩ p−1(I).
Then there exists T0 > 0 such that, if ρ ∈ K and t > T0,
(5.1) exp(tHp)(ρ) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn).
Proof. Let δ > 0. From the construction of C. Gérard and J. Sjöstrand [17], there exists a
function G(x, ξ) ∈ C∞(R2n) such that,
(HpG)(x, ξ) ≥ 0 for all (x, ξ) ∈ p−1(]
[),(5.2)
(HpG)(x, ξ) > 2E(1− δ) for |x| > R0 and p(x, ξ) = E ∈]
[,(5.3)
G(x, ξ) = x · ξ for |x| > R0.(5.4)
Let ρ ∈ K, and γ(t) = (x(t), ξ(t)) = exp(tHp)(ρ) be the corresponding Hamiltonian curve.
We distinguish between 2 cases:
1) For all t > 0, we have |x(t)| > R0.
Then G(γ(t)) > 2E(1 − δ)t+G(ρ) and, for t > T1 with T1 large enough,
(5.5) G(γ(t)) > 2 sup
x∈B(0,R0)
p(x,ξ)∈I
G(x, ξ).
20 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
By continuity, there exists a neighborhood U of γ such that, for all γ̃ ∈ U , we have
(5.6) G(γ̃(T1)) > sup
x∈B(0,R0)
p(x,ξ)∈I
G(x, ξ).
Since G is non-decreasing on γ̃(t), we have |x̃(t)| > R0 for all t > T1, and then
(5.7) G(γ̃(t)) > 2E(1 − δ)(t − T1) +G(γ̃(T1)) > 2E(1 − δ)t − C.
On the other hand, by uniformly finite propagation, we have |x̃(t)| <
2E(1+ δ)t+C. From
(5.7), we get |x̃(t)| > 1
t− C for all γ̃ ∈ U , and then |ξ̃(t)| =
2E + ot→∞(1). In particular,
the previous estimates gives
(5.8) |x(t)| > R/2,
(5.9) cos
x̃, ξ̃
(t) >
2E(1− δ)t− C
2E(1 + δ)t+ C)(
2E + ot→∞(1))
1 + δ
+ ot→∞(1) > 1− 3δ,
for t > T0 with T0 large enough but independent on γ̃ ∈ U . Thus, for t > T0 and γ̃ ∈ U , we
(5.10) γ̃(t) ∈ Γ+(R/2, d, σ+),
with σ+ = 1− 3δ.
2) There exist T2 > 0 such that |x(T2)| = R0.
Then there exists V a neighborhood of γ such that, for all γ̃ ∈ V, we have |x̃(T2)| < 2R0. Let
t > T2.
a) If |x̃(t)| ≤ R/2, then γ̃(t) ∈ B(0, R/2) × Rn.
b) Assume now |x̃(t)| > R/2. Denote by T3 (> T2) the last time (before t) such that
|x̃(T3)| = 2R0. Then
G(γ̃(t)) >2E(1 − δ)(t− T3) +G(γ̃(T3))(5.11)
>2E(1 − δ)(t− T3)− C,(5.12)
where C depend only on R0. On the other hand, the have |x̃(t)| <
2E(1 + δ)(t − T3) + C
(where the constant C depend only on R0). Then,
(5.13) t− T3 >
|x̃(t)|√
2E(1 + δ)
2E(1 + δ)
(5.14) |ξ̃(t)| =
2E + oR→∞(1),
x̃, ξ̃
(t) >
2E(1− δ)|x̃(t)|
|x̃(t)|(
2E(1 + δ))(
2E + oR→∞(1))
+O(R−1)
1 + δ
+ oR→∞(1) > 1− 2δ + oR→∞(1).(5.15)
So, if R is large enough, γ̃(t) ∈ Γ+(R/2, d, σ+), σ+ = 1− 3δ.
Then a) and b) imply that, for all γ̃ ∈ V and t > T0 := T2, we have
(5.16) γ̃(t) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 21
The lemma follows from (5.10), (5.16) and a compactness argument. �
Recall that the microsupport of g−(x)e
iψ−(x)/h ∈ C∞0 (Rn) is contained in Γ−(R−, d1, σ1).
Let ω−(x, ξ) ∈ A0 with ω− = 1 near Γ−(R−/2, d1, σ1) and supp(ω−) ⊂ Γ−(R−/3, d0, σ0).
Using the identity
(5.17) u− =
e−it(P−E)/h(g−e
iψ−/h)dt+R(E + i0, h)e−iT (P−E)/h(g−eiψ−/h),
and Proposition 4.1, Proposition 4.2 and Lemma 5.1, we get
(5.18) Oph(ω−)u− = Oph(ω−)
e−it(P−E)/h(g−e
iψ−/h)dt+O(h∞),
for some T > 0 large enough. In particular,
(5.19) MS(Oph(ω−)u−) ⊂ Λ−ω ∩ (B(0, R− + 1)× Rn).
5.2. Computation of u− along γ
Now we want to compute u− microlocally along a trajectory γ
k . We recall that γ
k is a
bicharacteristic curve (x−k (t), ξ
k (t)) such that (x
k (t), ξ
k (t)) → (0, 0) as t → +∞, and such
that, as t→ −∞,
(5.20)
|x−k (t)−
2E0ωt− z−k | → 0,
2E0ω| → 0.
If R− is large enough, a− solves (4.7) and (4.8) microlocally near γ
k ∩ MS(g−eiψ−/h). In
particular, microlocally near γ−k ∩ Γ−(R−/2, d1, σ1) ∩ (B(0, R−)× Rn), u− is given by (5.18)
e−it(P−E)/h([χ−, P ]a−e
iψ−/h)dt+O(h∞)
e−it(P−E)/h(χ−(P − E)a−eiψ−/h)dt
e−it(P−E)/h((P − E)χ−a−eiψ−/h)dt+O(h∞)
(P − E)e−it(P−E)/h(χ−a−eiψ−/h)dt+O(h∞)
=(P − E)R(E + i0, h)a−eiψ−/h +O(h∞)
iψ−/h +O(h∞).(5.21)
Now, using (5.21), and the fact that u− is a semiclassical distribution satisfying
(5.22) (P − E)u− = 0,
we can compute u− microlocally near γ
∩ B(0, R−) using Maslov’s theory (see [23] for
more details). Moreover, it is proved in Proposition C.1 (see also [5, Lemma 5.8]) that the
Lagrangian manifold Λ−ω has a nice projection with respect to x in a neighborhood of γ
close
to (0, 0). Then, in such a neighborhood, u− is given by
(5.23) u−(x) = a−(x, h)e
π/2eiψ−(x)/h,
22 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where ν−k denotes the Maslov index of γ
k , and ψ− satisfies the usual eikonal equation
(5.24) p(x,∇ψ−) = E0.
Here, to the contrary of (4.21), we have written E = E0 + zh with z = O(1), and we choose
to work with z in the amplitudes instead of the phases. As usual, we have
(5.25) ∂t(ψ−(x
(t))) = ∇ψ−(x−k (t)) · ∂tx
(t) = ∇ψ−(x−k (t)) · ξ
(t) = |ξ−
(t)|2,
so that
(5.26) ψ−(x
(t)) = ψ−(x
(s)) +
(u)|2du
We also have ψ−(x
k (s)) = (
2E0ωs+ z
k ) ·
2E0ω + o(1) as s→ −∞, and then
(5.27) ψ−(x
k (t)) = 2E0s+
|ξ−k (u)|
2du+ o(1), s→ −∞.
We have obtained in particular that
(5.28) ψ−(x
k (t)) =
|ξ−k (u)|
2−2E01u<0 du =
|ξ−k (u)|
2−V (x−k (u))+E0 sgn(u) du.
We turn to the computation of the symbol. The function a−(x, h) ∼
k=0 a−,k(x)h
satisfies the usual transport equations:
(5.29)
∇ψ− · ∇a−,0 +
(∆ψ− − 2iz)a−,0 = 0,
∇ψ− · ∇a−,k +
(∆ψ− − 2iz)a−,k = i
∆a−,k−1, k ≥ 1,
In particular, we get for the principal symbol
(5.30) ∂t(a−,0(x
(t))) = ∇a−,0(x−k (t)) · ξ
(t) = ∇a−,0(x−k (t)) · ∇ψ−(x
(t)),
so that,
(5.31) ∂t(a−,0(x
k (t))) = −
∆ψ−(x
k (t))− 2iz
a−,0(x
k (t))
and then
(5.32) a−,0(x
k (t)) = a−,0(x
k (s)) exp
∆ψ−(x−(u)) du + i(t− s)z
On the other hand, from [30, Lemma 4.3], based on Maslov theory, we have
(5.33) a−,0(x
k (t)) = (2E0)
1/4D−k (t)
−1/2eitz ,
where
(5.34) D−
(t) =
∣∣ det
∂x−(t, z, ω,E0)
∂(t, z)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 23
6. Computation of u− at the critical point
Now we use the results of [5] to get a representation of u− in a whole neighborhood of
the critical point. Indeed we saw already that (P − E)u− = 0 out of the support of g−, in
particular in a neighborhood of the critical point. First, we need to recall some terminology
of [18] and [5].
We recall from Section 2 that (µj)j≥0 is the strictly growing sequence of linear combinations
over N of the λj’s. Let u(t, x) be a function defined on [0,+∞[×U , U ⊂ Rm.
Definition 6.1. We say that u : [0,+∞[×U → R, a smooth function, is expandible, if, for
any N ∈ N, ε > 0, α, β ∈ N1+m,
(6.1) ∂αt ∂
u(t, x)−
uj(t, x)e
−µj t
e−(µN+1−ε)t
for a sequence of (uj)j smooth functions, which are polynomials in t. We shall write
u(t, x) ∼
uj(t, x)e
−µj t,
when (6.1) holds.
We say that f(t, x) = Õ(e−µt) if for all α, β ∈ N1+m and ε > 0 we have
(6.2) ∂αt ∂
xf(t, x) = O(e−(µ−ε)t).
Definition 6.2. We say that u(t, x, h), a smooth function, is of class SA,B if, for any ε > 0,
α, β ∈ N1+m,
(6.3) ∂αt ∂
xu(t, x, h) = O
hAe−(B−ε)t
Let S∞,B =
A SA,B. We say that u(t, x, h) is a classical expandible function of order (A,B),
if, for any K ∈ N,
(6.4) u(t, x, h) −
uk(t, x)h
k ∈ SK+1,B,
for a sequence of (uk)k expandible functions. We shall write
u(t, x, h) ∼
uk(t, x)h
in that case.
Since the intersection between Λ−ω and Λ− is transverse along the trajectories γ
k ), and
since g−1 (z
k ) 6= 0, Theorem 2.1 and Theorem 5.4 of [5] implies that one can write, microlocally
near (0, 0),
(6.5) u− =
∫ N−∑
αk(t, x, h)eiϕ
k(t,x)/hdt,
24 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where the αk(t, x, h)’s are classical expandible functions in S0,2ReΣ(E):
(6.6)
αk(t, x, h) ∼
αkm(t, x)h
αkm(t, x) ∼
αkm,j(t, x)e
−2(Σ(E)+µj )t,
and where the αkm,j(t, x)’s are polynomial with respect to t. We recall from (2.26) that, for
E = E0 + hz,
(6.7) Σ(E) =
− iz.
Following line by line Section 6 of [5], we obtain (see [5, (6.26)])
αk0,0(0) = e
iπ/4(2λ1)
3/2e−iν
π/2|g(γ−k )|(D
−1/2(2E0)
1/4.(6.8)
Notice that from (5.32) and Proposition C.1, we have 0 < D−
< +∞.
From [5, Section 5], we recall that the phases ϕk(t, x) satisfies the eikonal equation
(6.9) ∂tϕ
k + p(x,∇xϕk) = E0,
and that they have the asymptotic expansion
(6.10) ϕk(t, x) ∼
ϕkj,m(x)t
me−µjt,
with Mkj < +∞. In the following, we denote
(6.11) ϕkj (t, x) =
ϕkj,m(x)t
and the first ϕkj ’s are of the form
ϕk0(t, x) =ϕ+(x) + ck(6.12)
ϕk1(t, x) =− 2λ1g−(z−k ) · x+O(x
2),(6.13)
where ck ∈ R is the constant depending on k given by
(6.14) ck = “ψ−(0)” = lim
k (t)) = S
thanks to (5.28) (see also [5, Lemma 5.10]). Moreover ϕ+ is the generating function of the
outgoing stable Lagrangian manifold Λ+ with ϕ+(0) = 0. We have
(6.15) ϕ+(x) =
x2j +O(x3).
The fact that ϕk1(t, x) does not depend on t and the expression (6.13) follows also from
Corollary 6.6 and (6.109).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 25
6.1. Study of the transport equations for the phases.
Now, we examine the equations satisfied by the functions ϕkj (t, x), defined in (6.10), for
the integers j ≤ ̂ (recall that ̂ is defined by µb = 2λ1). For clearer notations, we omit the
superscript k until further notice.
Let us recall that the function ϕ(t, x) satisfies the eikonal equation (6.9), which implies
(see (6.10))
(6.16)
e−µjtϕj,m(x)(−µjtm+mtm−1)+
∇ϕj,m(x)tme−µj t
+V (x) ∼ E0,
and then
e−µjtϕj,m(x)(−µjtm +mtm−1) +
∇ϕj,m∇ϕe,em(x)e−(µj+µe)ttm+ em
+V (x) ∼ E0.(6.17)
When µj < 2λ1, the double product of the previous formula provides a term of the form e
if and only if µj = 0 or µe = 0. In particular, the term in e
−µjt in (6.17) gives
(6.18)
ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) ·
∇ϕj,m(x)tm = 0.
When µj = 2λ1, one gets also a term in e
−2λ1t for µj = µe = λ1 and then
ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) ·
∇ϕj,m(x)tm
tm+ em∇ϕ1,m(x)∇ϕ1, em(x) = 0.(6.19)
We denote
(6.20) L = ∇ϕ+(x) · ∇
the vector field that appears in (6.18) and (6.19). We set also L0 =
j λjxj∂j its linear part
at x = 0, and we begin with the study of the solution of
(6.21) (L− µ)f = g,
with µ ∈ R and f , g ∈ C∞(Rn). First of all, we show that it is sufficient to solve (6.21) for
formal series.
Proposition 6.3. Let g ∈ C∞(Rn) and g0 the the Taylor expansion of g at 0. For each
formal series f0 such that (L−µ)f0 = g0, there exists one and only one function f ∈ C∞(Rn)
defined near 0 such that f = f0 +O(x∞) and
(6.22) (L− µ)f = g,
near 0.
26 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Proof. Let f̃0 be a C
∞ function having f0 has Taylor expansion at 0. With the notation
f = f̃0 + r, the problem (6.22) is equivalent to find r = O(x∞) with
(6.23) (L− µ)r = g − (L− µ)f̃0 = r̃,
where r̃ ∈ C∞ has g0 − (L− µ)f0 = 0 as Taylor expansion at 0. Let y(t, x) be the solution of
(6.24)
∂ty(t, x) = ∇ϕ+(y(t, x)),
y(0, x) = x.
Thus, (6.23) is equivalent to
(6.25) r(x) =
e−µsr̃(y(s, x))ds + e−µtr(y(t, x)).
Since r(x), r̃(x) = O(x∞) and y(s, x) = O(eλ1t|x|) for t < 0, the functions e−µtr(y(t, x)),
e−µtr̃(y(t, x)) are O(eNt) as t→ −∞ for all N > 0. Then
(6.26) r(x) =
e−µsr̃(y(s, x))ds,
and r(x) = O(x∞). The uniqueness follows and it is enough to prove that r given by (6.26)
is C∞. We have
(6.27) ∂t(∇xy) = (∇2xϕ+(y))(∇xy),
and since ∇2xϕ+ is bounded, there exists C > 0 such that
(6.28) |∇xy(t, x)| . e−Ct,
has t → −∞. Then, e−µs(∇r̃)(y(s, x))(∂jy(t, x)) = O(eNt) as t → −∞ for all N > 0 and
∂jr(x) =
−µs(∇r̃)(y(s, x))(∂jy(t, x))ds. The derivatives of order greater than 1 can be
treated the same way. �
We denote
(6.29) Lµ = L− µ : CJxK → CJxK,
where we use the standard notation CJxK for formal series, and CpJxK for formal series of
degree ≥ p. We notice that
(6.30) Lµx
α = (L0 − µ)xα + C|α|+1JxK = (λ · α− µ)xα +C|α|+1JxK.
Recall that Iℓ(µ) has been defined in (2.20). The number of elements in Iℓ(µ) will be denoted
(6.31) nℓ(µ) = #Iℓ(µ).
One has for example n2(µ) =
n1(µ)(n1(µ)+1)
Proposition 6.4. Suppose µ ∈]0, 2λ1[. With the above notations, one has KerLµ⊕ ImLµ =
CJxK. More precisely:
i) The kernel of Lµ has dimension n1(µ), and one can find a basis (Ej1 , . . . , Ejn1(µ)
KerLµ such that Ej(x) = xj + C2JxK, j ∈ I1(µ).
ii) A formal series F = F0 +
Fjxj + C2JxK belongs to ImLµ if and only if Fj = 0 for
all j ∈ I1(µ).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 27
Remark 6.5. Thanks to Propostion 6.3, the same result is true for germs of C∞ functions
at 0. Notice that when µ 6= µj for all j, Lµ is invertible.
Proof. For a given F =
α Fαx
α ∈ CJxK, we look for solutions E =
α ∈ CJxK to the
equation
(6.32) Lµ
The calculus of the term of order x0 in (6.32) leads to the equation
(6.33) E0 = −
With this value for E0, (6.32) becomes, using again (6.30),
(6.34)
|α|=1
(λ · α− µ)Eαxα =
|α|=1
α + C2JxK.
We have two cases:
If α /∈ I1(µ), one should have
(6.35) Eα =
λ · α− µ.
If α ∈ I1(µ), the formula (6.34) becomes Fα = 0. In that case, the corresponding Eα can
be chosen arbitrarily.
Now suppose that the Eα are fixed for any |α| ≤ n− 1 (with n ≥ 2), and such that
(6.36) Lµ
|α|≤n−1
α + CnJxK.
We can write (6.32) as
(6.37) Lµ
|α|=n
α − Lµ
|α|≤n−1
+ Cn+1JxK,
or, using again (6.30),
(6.38)
|α|=n
(λ · α− µ)Eαxα =
|α|≤n
α − Lµ
|α|≤n−1
+Cn+1JxK.
Since |α| ≥ 2, one has λ · α ≥ 2λ1 > µ, so that (6.38) determines by induction all the Eα’s
for |α| = n in a unique way. �
Corollary 6.6. If j < ̂, the function ϕj(t, x) does not depend on t, i.e. we have Mj = 0.
Proof. Suppose that Mj ≥ 1, then (6.18) gives the system
(6.39)
(L− µj)ϕj,Mj = 0,
(L− µj)ϕj,Mj−1 = −Mjϕj,Mj ,
with ϕj,Mj 6= 0. But this would imply that ϕj,Mj ∈ KerLµ ∩ ImLµ, a contradiction. �
28 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
As a consequence, for j < ̂, the equation (6.18) on ϕj reduces to
(6.40) (L− µj)ϕj,0 = 0,
and, from Proposition 6.4, we get that
(6.41) ϕj(t, x) = ϕj,0(x) =
k∈I1(µ)
dj,kxk +O(x2).
Now we pass to the case j = ̂, and we study (6.19). First of all, we have seen that ϕ1 does
not depend on t, so that this equation can be written
(6.42)
ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+ ·
∇ϕj,m(x)tm +
∣∣∇ϕ1(x)
∣∣2 = 0.
As for the study of (6.18), we begin with that of (6.21), now in the case where µ = 2λ1.
We denote Ψ : Rn1(2λ1) −→ Rn2(λ1) the linear map given by
(6.43) Ψ(Eβ1 , . . . , Eβn1(2λ1)
β∈I1(2λ1)
∂α(L− µ)xβ
α∈I2(λ1)
and we set
(6.44) n(Ψ) = dimKerΨ.
Recalling that L = ∇ϕ+(x) · ∇, we see that
(6.45) Ψ(Eβ1 , . . . , Eβn1(2λ1)
β∈I1(2λ1)
∂α∂βϕ+(0)
α∈I2(λ1)
More generally, for any |α| = 2, we denote
(6.46) Ψα((Eβ)β∈I1(2λ1)) =
β∈I1(2λ1)
∂α∂βϕ+(0)
Then, at the level of formal series, we have the
Proposition 6.7. Suppose µ = 2λ1. Then
i) KerLµ has dimension n2(λ1) + n(Ψ).
ii) A formal series F =
α Fαx
α belongs to ImLµ if and only if
∀α ∈ I1(2λ1), Fα = 0,(6.47)
|β|=1
β /∈I1(2λ1)
∂β∂αϕ+(0)
2λ1 − λ · β
α∈I2(λ1)
∈ ImΨ.(6.48)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 29
iii) If F ∈ ImLµ, any formal series E =
α with LµE = F satisfies
F0,(6.49)
λ · α− 2λ1
Fα, for α ∈ I1 \ I1(2λ1),(6.50)
β∈I1(2λ1)
|β|=1
β /∈I1(2λ1)
∂β∂αϕ+(0)
2λ1 − λ · β
α∈I2(λ1)
.(6.51)
Moreover for α ∈ I2 \ I2(λ1), one has
(6.52) Eα =
λ · α− 2λ1
Fα −Ψα((Eβ)β∈I1(2λ1)) +
|β|=1
β /∈I1(2λ1)
2λ1 − λ · β
∂α+βϕ+(0)
Last, E is completely determined by F and a choice of the Eα for |α| ≤ 2 such that
(6.49)– (6.52) are satisfied.
iv) KerLµ ∩ Im(Lµ)2 = {0}.
Proof. For a given F =
α Fαx
α we look for a E =
α such that L2λ1E = F . First of
all, we must have
(6.53) E0 = −
When this is true, we get
(6.54)
|α|=1
Eα(L0 − 2λ1)xα =
|α|=1
Fα(L− 2λ1)xα + C2JxK,
and we obtain as necessary condition that Fα = 0 for any α ∈ I1(2λ1). So far, the Eα for
α ∈ I1(2λ1) can be chosen arbitrarily, and we must have
(6.55) Eα =
λ · α− 2λ1
, α ∈ I2 \ I1(2λ1).
We suppose that (6.53) and (6.55) hold. Then we should have
(6.56)
|α|=2
Eα(L0−2λ1)xα =
|α|=2
|α|=1
α/∈I1(2λ1)
|α|=1
Eα(L−2λ1)xα
+C3JxK.
Notice that the second term in the R.H.S of (6.56) belongs to C2JxK thanks to (6.55). Again,
we have to cases:
• When α ∈ I2(λ1), the corresponding Eα can be chosen arbitrarily, but one must have
|β|=1
∂α(L− 2λ1)xβ
|x=0(6.57)
=Ψα((Eβ)β∈I1(2λ1)) +
|β|=1
β /∈I1(2λ1)
∂α+βϕ+(0)
,(6.58)
30 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
and this, with (6.55), gives (6.51).
• When |α| = 2, α /∈ I2(λ1), one obtains
λ · α− 2λ1
|β|=1
∂α(L− 2λ1)xβ
λ · α− 2λ1
Fα −Ψα((Eβ)β∈I1(2λ1))−
|β|=1
β /∈I1(2λ1)
∂α+βϕ+(0)
,(6.59)
and this, with (6.55), gives (6.52).
Now suppose that (6.53), (6.55), (6.57) and (6.59) hold, and that we have chosen a value
for the free variables Eα for α ∈ I1(2λ1)∪I2(λ1). Thanks to the fact that λ ·α 6= 2λ1 for any
α ∈ Nn with |α| = 3, we see as in the proof of Propostion 6.4, that the equation (6.54) has a
unique solution, and the points (i), (ii) and (iii) follows easily.
We prove the last point of the proposition, and we suppose that
(6.60) E =
α ∈ KerLµ ∩ Im(Lµ)2.
First, we have E ∈ KerLµ ∩ ImLµ. Thus, E0 = 0 by (6.49), Eα = 0 for α ∈ I1(2λ1) by
(6.47), and Eα = 0 for α ∈ I1 \ I1(2λ1) by (6.50). Last, since LµE = 0, we also have Eα = 0
for α ∈ I2 \ I2(λ1), and finally,
(6.61) E =
α∈I2(λ1)
α + C3JxK.
Moreover, one can write E = LµG for some G ∈ ImLµ. Since E0 = 0, we must have G0 = 0.
Since G ∈ ImLµ, by (6.47), we have Gα = 0 for α ∈ I1(2λ1). Finally, since Eα = 0 for
|α| = 1, α /∈ I1(2λ1), the same is true for the corresponding Gα, and
(6.62) G =
|α|≥2
Then, since Lµx
α = 0+C3[x] for α ∈ I2(λ1), we obtain Eα = 0 for α ∈ I2(λ1). As above, we
then get that, for |α| ≥ 3, Eα = 0, and this ends the proof. �
Corollary 6.8. We always have Mb ≤ 2. If, in addition, λk 6= 2λ1 for all k ∈ {1, . . . , n}, then
Mb ≤ 1.
Proof. Suppose that Mb ≥ 3. Then (6.42) gives
(L− µb)ϕb,Mb = 0(6.63)
(L− µb)ϕb,Mb −1 = −Mbϕb,Mb(6.64)
(L− µb)ϕb,Mb −2 = −(Mb − 1)ϕb,Mb −1,(6.65)
with ϕb,Mb 6= 0. Notice that we have used the fact that Mb − 2 > 0 in (6.65). But this gives
ϕb,Mb ∈ Ker(L − µb) and (L − µb)2ϕb,Mb−2 = Mb(Mb − 1)ϕb,Mb , so that ϕb,Mb ∈ Im(L− µb)2.
This contradicts point (iv) of Proposition 6.7.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 31
Now we suppose that λk 6= 2λ1 for all k ∈ {1, . . . n}, that is I1(2λ1) = ∅, and that Mb = 2.
Then (6.42) gives
(L− µb)ϕb,Mb = 0(6.66)
(L− µb)ϕb,Mb −1 = −Mb ϕb,Mb(6.67)
with ϕb,Mb 6= 0. Therefore we have ϕb,Mb ∈ KerLµb ∩ ImLµb , and we get the same conclusion
as in (6.61): ϕb,Mb(x) = O(x2). Then, we write
(6.68) ϕb,Mb = (L− µb)g,
and we see, as in (6.62), that g = O(x2), here because I1(2λ1) = ∅. Finally, we conclude also
that ϕb,Mb = 0, a contradiction. �
6.2. Taylor expansions of ϕ+ and ϕ
Now we compute the Taylor expansions of the leading terms with respect to t, of the phase
functions ϕ(t, x) = ϕk(t, x).
Lemma 6.9. The smooth function ϕ+(x) =
x2j +O(x3) satisfies
(6.69) ∂αϕ+(0) = −
λ · α∂
αV (0),
for |α| = 3, and
(6.70) ∂αϕ+(0) = −
2(λ · α)
β,γ∈I2
α=β+γ
β! γ!
βV (0)
λj + λ · β
γV (0)
λj + λ · γ
λ · α∂
αV (0),
for |α| = 4, where α, β, γ ∈ Nn are multi-indices.
Proof. The smooth function x 7→ ϕ+(x) is defined in a neighborhood of 0, and it is charac-
terized (up to a constant: we have chosen ϕ+(0) = 0) by
(6.71)
p(x,∇ϕ+(x)) =
|∇ϕ+(x)|2 + V (x) = 0
∇ϕ+(x) = (λjxj)j=1,...,n +O(x
The Taylor expansion of ϕ+ at x = 0 is
(6.72) ϕ+(x) =
x2j +
|α|=3,4
∂αϕ+(0)
xα +O(x5),
and we have
(6.73) ∂jϕ+(x) = λjxj +
|α|=3,4
∂αϕ+(0)
xα−1j +O(x4).
32 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Therefore
|∇ϕ+(x)|2 =
j + 2
|α|=3
)∂αϕ+(0)
xα + 2
|α|=4
)∂αϕ+(0)
|α|=3
∂αϕ+(0)
xα−1j
+O(x5).(6.74)
Let us compute further the last term in (6.74):
|α|=3
∂αϕ+(0)
xα−1j
|β|,|γ|=3
∂βϕ+(0)
∂γϕ+(0)
xβ+γ−21j
|α|=4
α=β+γ
|β|,|γ|=2
βϕ+(0)
γϕ+(0)
·(6.75)
Writing the Taylor expansion of V at x = 0 as
(6.76) V (x) =
x2j +
|α|=3,4
∂αV (0)
xα +O(x5),
and using the eikonal equation (6.71), we obtain first, for any α ∈ Nn with |α| = 3,
(6.77) ∂αϕ+(0) = −
λ · α∂
αV (0).
Then, (6.74) and (6.75) give
(6.78) ∂αϕ+(0) = −
λ · α∂
αV (0) − 1
2(λ · α)
β,γ∈I2
α=β+γ
βV (0)
λj + λ · β
γV (0)
λj + λ · γ
for |α| = 4. �
Now we pass to the function ϕ1. This function is a solution, in a neighborhood of x = 0,
of the transport equation
(6.79) Lϕ1(x) = λ1ϕ1(x),
where L is given in (6.20).
Lemma 6.10. The C∞ function ϕ1(x) = −2λ1g−1 (z
) · x+O(x2) satisfies
(6.80) ∂αϕ1(0) =
2λ1α!
(λ1 − λ · α)(λ1 + λ · α)
αV (0)
g−1 (z
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 33
for |α| = 2, and
∂αϕ1(0) =−
λ1 − λ · α
k∈I1(λ1),j∈I1
β,γ∈I2
α+1j=β+γ
βV (0)
λj + λ · β
γV (0)
(λ1 − λ · γ)(λ1 + λ · γ)
g−1 (z
(λ1 − λ · α)(λ1 + λ · α)
k∈I1,j∈I1(λ1)
β,γ∈I2
1j+α=β+γ
(α+ 1j)!
βV (0)
λk + λ · β
γV (0)
λk + λ · γ
g−1 (z
(λ1 − λ · α)(λ1 + λ · α)
j∈I1(λ1)
αV (0)
g−1 (z
.(6.81)
for |α| = 3.
Proof. We write
(6.82) ϕ1(x) =
ajxj +
|α|=2,3
α +O(x4),
and Lemma 6.9 together with (6.73) give all the coefficients in the expansion
(6.83) ∇ϕ+(x) =
λjxj +
|α|=2,3
Aj,αx
α +O(x4)
j=1,...,n
In fact, we have
(6.84) Aj,α =
∂α+1jϕ+(0)
and aα =
∂αϕ1(0)
We get
Lϕ1(x) =
∂jϕ+(x)∂jϕ1(x)
ajλjxj +
|α|=2
αjλjaα + ajAj,α
|α|=3
αjλjaαx
|β|=|γ|=2
Aj,βγjaγx
β+γ−1j +
|α|=3
ajAj,αx
+O(x4)
ajλjxj +
|α|=2
λ · α aα +
Aj,αaj
|α|=3
λ · α aα +
α=β+γ−1j
|β|,|γ|=2
Aj,βγjaγ + ajAj,α
xα +O(x4).(6.85)
34 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Thus, (6.79) gives, for all α ∈ Nn with |α| = 2,
(6.86) aα =
λ1 − λ · α
Aj,αaj ,
and, for all α ∈ Nn with |α| = 3,
(6.87) aα =
λ1 − λ · α
β,γ∈I2
α+1j=β+γ
γjAj,βaγ + ajAj,α
Then, the lemma follows from (6.84). �
6.3. Asymptotics near the critical point for the trajectories.
The informations obtained so far are not sufficient for the computation of the ϕj ’s. We
shall obtain here some more knowledge by studying the behaviour of the incoming trajectory
γ−(t) as t → +∞. We recall from [18, Section 3] (see also [5, Section 5]), that the curve
γ−(t) = (x−(t), ξ−(t)) ∈ Λ− ∩ Λ−ω satisfy, in the sense of expandible functions,
(6.88) γ−(t) =
M ′j∑
γ−j,mt
me−µjt,
Notice that we continue to omit the subscript k for γ−k = (x
k , ξ
k ), z
k , . . . Writing also
(6.89) x−(t) ∼
g−j,m(t, z−)e
−µj t, g−j (z
−, t) =
M ′j∑
g−j,m(z
−)tm,
for some integers M ′j, we know that g
−) = g−1,0(z
−) 6= 0. Since ξ−(t) = ∂tx−(t), we have
(6.90) ξ−(t) ∼
M ′j∑
g−j,m(z
−)(−µjtm +mtm−1)e−µjt.
Proposition 6.11. If j < ̂, then M ′j = 0. We also have M
≤ 1, and M ′
= 0 when
I1(2λ1) 6= ∅. Moreover
(6.91) (g−
|α|=2
∂α+βV (0)
(g−1 (z
−))α for β ∈ I1(2λ1),
0 for β /∈ I1(2λ1).
and, for |β| = 1, β /∈ I1(2λ1),
(6.92) (g−
(2λ1 + λ · β)(2λ1 − λ · β)
|α|=2
∂α+βV (0)
(g−1 (z
−))α.
Proof. First of all, since ∂tγ
−(t) = Hp(γ
−(t)), we can write
(6.93) ∂tγ
−(t) = Fp(γ
−(t)) +O(t2M ′1e−2λ1t),
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 35
where
(6.94) Fp = d(0,0)Hp =
, Λ2 = diag(λ21, . . . , λ
We obtain
(6.95)
1≤j<b
M ′j∑
(Fp + µj)γ
1≤j<b
M ′j∑
γ−j,mmt
m−1e−µj t.
Now suppose j < ̂ and M ′j ≥ 1. We get, for this j, for some γ
j,M ′j
6= 0,
(6.96)
(Fp + µj)γ
j,M ′j
(Fp + µj)γ
j,M ′j−1
=M ′jγ
j,M ′j
so that Ker(Fp + µj) ∩ Im(Fp + µj) 6= {0}. Since Fp is a diagonizable matrix, this can easily
be seen to be a contradiction.
Now we pass to the study of M ′
. So far we have obtained that
(6.97) γ−(t) =
1≤j<b
γ−j e
−µjt +
tme−2λ1t +O(tCe−µb+1t),
and we can write
(6.98) Hp(x, ξ) =
|α|=2
∂α∇V (0)
xα +O(x3)
 .
Thus we have
(6.99) Hp(γ
−(t)) = Fp
γ−j e
−µjt +
tme−2λ1t
+ e−2λ1tA(γ−1 ) +O(e
−(2λ1+ε)t),
where, noticing that µj + µj′ = 2λ1 if and only if j = j
′ = 1,
(6.100) A(γ−1 ) =
|α|=2
∂α∇V (0)
(g−1 )
 .
For the terms of order e−2λ1t, we have, since ∂tγ
−(t) = Hp(γ
−(t)),
(6.101) (Fp + 2λ1)
mtm−1 −A(γ−1 ).
Thus, if we suppose that M ′
≥ 2, we obtain
(6.102)
(Fp + 2λ1)γ
b,M ′
(Fp + 2λ1)γ
b,M ′
b,M ′
36 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Then again we have γ−
b,M ′
∈ Ker(Fp + 2λ1) ∩ Im(Fp + 2λ1), a contradiction.
Eventually, if λj 6= 2λ1 for all j, then Ker(Fp + 2λ1) = {0}. Therefore, if we suppose that
b = 1, we see that γb,1 6= 0 satisfies the first equation in (6.102) and we get a contradiction.
Now we compute γ−
(t) = γ−
t+ γ−
. We have
(6.103)
(Fp + 2λ1)γ
(Fp + 2λ1)γ
−A(γ−1 ),
and we see that γ−
= Πγ−
= ΠA(γ−1 ), where Π is the projection on the eigenspace of
Fp associated to −2λ1. We denote by ej = (δi,j ⊗ 0)i=1,...,n and εj = (0 ⊗ δi,j)i=1,...,n for
j = 1, . . . , n, so that (e1, . . . en, ε1, . . . , εn) is the canonical basis of R
2n = T(0,0)T
n. Then it
is easy to check that , for all j, v±j = ej±λj1εj is an eigenvector of Fp for the eigenvalue ±λj .
In the basis {e1, ε1, . . . , en, εn} the projector Π is block diagonal and, if Kj = Vect(ej , εj), we
(6.104) Π|Kj
1/2 −1/4λ1
−λ1 1/2
for j ∈ I1(2λ1),
0 for j /∈ I1(2λ1).
Therefore, we obtain
(6.105) (g−
|α|=2
∂β∂αV (0)
(g−1 (z
−))α for β ∈ I1(2λ1),
0 for β /∈ I1(2λ1).
Now suppose that k /∈ I1(2λ1). Then the second equality in (6.103) restricted to Kk gives
(6.106)
2λ1 1
λ2k 2λ1
Πkγb,0 = −ΠkA(γ−1 ),
where Πk denotes the projection onto Kk. Solving this system, one gets
(6.107) (g−
4λ21 − λ2k
ΠxΠkA(γ
and, together with (6.100), this ends the proof of Proposition 6.11. �
6.4. Computation of the ϕkj ’s.
Here we compute the ϕkj ’s for j ≤ ̂. We still omit the superscript k. From [5], we know
that ξ−(t) = ∇xϕ
t, x−(t)
, so that, using (6.41),
ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t +
2≤j<b
∇ϕj(0)e−µj t
+∇ϕb,2(0)t2e−2λ1t +∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t).(6.108)
Since ϕ+ = −ϕ− and ξ− ∈ Λ−, we have ∇ϕ+(x−(t)) = −ξ−(t), and we obtain first, by (6.90),
(6.109) ∇ϕj(0) = −2µjg−j (z
for 1 ≤ j < ̂.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 37
Now we study ϕb(t, x) = ϕb,0(x) + tϕb,1(x) + t
2ϕb,2(x) when I1(2λ1) 6= ∅. It follows from
(6.108) that we have
(6.110)
− 4λ1g−b,1(z
−) = ∇ϕb,1(0),
− 4λ1g−b,0(z
−) + 2g−
(z−) = ∇ϕb,0(0) +∇2ϕ1(0)g−1 (z
On the other hand, we have seen that, by (6.19), the functions ϕb,2, ϕb,1 and ϕb,0 satisfy
(6.111)
(L− 2λ1)ϕb,2 = 0,
(L− 2λ1)ϕb,1 = −2ϕb,2,
(L− 2λ1)ϕb,2 = −ϕb,1 −
|∇ϕ1(0)|2.
In particular ϕb,2 ∈ Ker(L− 2λ1) ∩ Im(L− 2λ1) so that (see (6.61)),
(6.112) ϕb,2(x) =
α∈I2(λ1)
c2,αx
α +O(x3).
Going back to (6.108), we notice that we obtain now
ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t +
2≤j<b
∇ϕj(0)e−µj t
∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t),(6.113)
and this equality is consistent with Proposition 6.11.
Then, (6.49) and (6.50) give
(6.114) ϕb,1(x) =
α∈I1(2λ1)
c1,αx
|α|=2
c1,αx
α +O(x3),
and, by (6.51), we have
(6.115) Ψ((c1,β)β∈I1(2λ1)) = (−2c2,α)α∈I2(λ1).
By (6.52), we also have for |α| = 2, α /∈ I2(λ1),
(6.116) c1,α =
2λ1 − λ · α
β∈I1(2λ1)
∂α+βϕ+(0)
c1,β .
The function ϕb,0(x) =
|α|≤2 c0,αx
α +O(x3) satisfies (see (6.42))
(6.117) (L− 2λ1)ϕb,0 = −ϕb,1 −
∣∣∇ϕ1(x)
First of all, the compatibility condition (6.47) gives
(6.118) ∀α ∈ I1(2λ1), c1,α = −∇ϕ1(0) · ∂α∇ϕ1(0),
so that in particular, by (6.115), the function ϕb,2 is known up to O(x3) terms:
(6.119) ∀α ∈ I2(λ1), c2,α =
β∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0),
38 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
(6.120) ∀α /∈ I2(λ1), |α| = 2, c1,α = −
2λ1 − λ · α
β∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0).
Now (6.49) and (6.50) give
(6.121) c0,0 = ϕb,0(0) =
|∇ϕ1(0)|2,
(6.122) ∀α /∈ I1(2λ1), |α| = 1, c0,α =
2λ1 − λ · α
∇ϕ1(0) · ∂α∇ϕ1(0).
From the other compatibility condition (6.48), we know that
c1,α +
∇ϕ1(0) · ∂α∇ϕ1(0) +
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0)
|β|=1
β/∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0)
2λ1 − λ · β
α∈I2(λ1)
∈ ImΨ,(6.123)
and, from (6.51), we obtain a relation between the (c0,β)β∈I1(2λ1) and the (c1,α)α∈I2(λ1),
namely
∀α ∈ I2(λ1), c1,α =−
∂α∇ϕ1(0) · ∇ϕ1(0)−
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0)
β∈I1(2λ1)
∂α+βϕ+(0)
c0,β −
|β|=1
β/∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0)
2λ1 − λ · β
·(6.124)
Using the second equation in (6.110), we obtain, for |β| = 1,
(6.125) c0,β = −4λ1(g−b,0(z
−))β + 2(g−
(z−))β − ∂β∇ϕ1(0) · g−1 (z
At this point, we have computed the functions ϕb,1(x) and ϕb,2(x) up to O(x3), in terms
of derivatives of ϕ+ and ϕ1, and of the g
(z−). We shall now use the expressions we have
obtained in Section 6.2 and in Section 6.3 to give these functions in terms of g−1 and of
derivatives of V only.
First of all, by (6.112), (6.119), Lemma (6.9) and Lemma (6.10), we obtain
ϕb,2(x) =−
γ∈I1(2λ1)
α,β∈I2(λ1)
∂β+γV (0)
(g−1 (z
∂α+γV (0)
+O(x3).(6.126)
Then we have
(6.127) ϕb,1(x) = −4λ1g−b,1(z
−) · x+
α∈I2(λ1)
c1,αx
|α|=2
α/∈I2(λ1)
c1,αx
α +O(x3),
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 39
where the c1,α are given by (6.124) and (6.125) for α ∈ I2(λ1), and by (6.120) for α /∈ I2(λ1).
• For |α| = 2, α /∈ I2(λ1), we obtain by (6.116), Lemma 6.9 and Lemma 6.10,
c1,α =
(2λ1 + λ · α)(2λ1 − λ · α)
β∈I1(2λ1)
∂α+βV (0)
(λ1 + λj)(3λ1 + λj)
β∇V (0) · g−1 (z−)(g
−))j .(6.128)
Since (g−1 (z
−))j = 0 but for j ∈ I1(λ1), we get, changing notations a bit,
(6.129) c1,α =
(2λ1 + λ · α)(2λ1 − λ · α)
γ∈I1(2λ1)
β∈I2(λ1)
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))β.
• Now we compute c1,α for α ∈ I2(λ1).
For the last term in the R.H.S. of (6.124), we obtain
|β|=1
β /∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0)
2λ1 − λ · β
γ∈I1\I1(2λ1)
β∈I2(λ1)
(2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))β .(6.130)
Using (6.91) and (6.125), we have also
β∈I1(2λ1)
∂α+βϕ+(0)
c0,β =
γ∈I1(2λ1)
∂α+γV (0)
(z−))γ +
γ∈I1(2λ1)
β∈I2(λ1)
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))β .(6.131)
40 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
We pass to the computation of − 1
∂α∇ϕ1(0) · ∇ϕ1(0) for α ∈ I2(λ1). We obtain
∂α∇ϕ1(0) · ∇ϕ1(0) = −
β∈I2(λ1)
∂α+βV (0)
(g−1 (z
j,p,k=1
β,γ∈I2
β+γ=α+1p+1j
((α+ 1p)j + 1)(αp + 1)
(λk + λ · β)(λk + λ · γ)
∂β+1kV (0)
∂γ+1kV (0)
(g−1 (z
−))j(g
+ 2λ1
j,p,k=1
β,γ∈I2
β+γ=α+1p+1j
(αp + 1)γj
(λ1 − λ · γ)(λ1 + λ · γ)(λj + λ · β)
β+1jV (0)
∂γ+1kV (0)
(g−1 (z
−))k(g
= I + II + III.
(6.132)
Writing δ = 1j + 1p, we get
(6.133) II = −1
β,γ,δ∈I2
β+γ=α+δ
(α+ δ)!
(λk + λ · β)(λk + λ · γ)
∂β+1kV (0)
∂γ+1kV (0)
(g−1 (z
α! δ!
Since δ ∈ I2(λ1) (otherwise (g−1 (z−))δ = 0), we have β, γ ∈ I2(λ1) and, changing notations a
(6.134) II = −1
β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
In the last term III, we can suppose that γ = 1j+1q for some q ∈ {1, . . . , n}. Then γj = γ!
and, writing β = 1a + 1b we have
III = λ1
j,k,p=1
(αp + 1)(g
−))k(g
a,b,q∈I1
1a+1b+1q=α+1p
(αp + 1)
(λ1 − λj − λq)(λ1 + λj + λq)(λj + λa + λb)
∂j,a,bV (0)∂j,q,kV (0).(6.135)
Since α ∈ I2(λ1) and 1p ∈ I1(λ1) (otherwise (g−1 (z−))p = 0), we have 1a, 1b, 1q ∈ I1(λ1) so
that we can write
(6.136)
III = −
j,k,p=1
(αp + 1)
λj(2λ1 + λj)2
(g−1 (z
−))k(g
a,b,q∈I1
1a+1b+1q=α+1p
∂j,a,bV (0)∂j,q,kV (0).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 41
Now it is easy to check, noticing that (α+ 1p)k ∈ {1, 2, 3, 4} and examining each case, that
(6.137)
a,b,q∈I1
1a+1b+1q=α+1p
∂j,a,bV (0)∂j,q,kV (0) =
(α+ 1p)k
a,b,c,d∈I1
1a+1b+1c+1d=α+1p+1k
∂j,a,bV (0)∂j,c,dV (0).
Therefore, we have
III = −1
j,k,p=1
(α + 1p + 1k)!
λj(2λ1 + λj)2
(g−1 (z
−))k(g
a,b,c,d∈I1
1a+1b+1c+1d=α+1p+1k
∂j,a,bV (0)∂j,c,dV (0).(6.138)
Eventually, setting β = 1p + 1k, γ = 1a + 1b and δ = 1c + 1d, we get
(6.139)
III = −
β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
λj(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
We are left with the computation of
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0) = −
β,γ∈I1(λ1)
β+γ=α
βϕ1(0) · ∂j∂γϕ1(0)
λ2j(2λ1 + λj)
β,γ∈I1(λ1)
β+γ=α
k,ℓ=1
∂j∂k∂
βV (0)(g−1 (z
−))k∂j∂ℓ∂
γV (0)(g−1 (z
−))ℓ.(6.140)
At this point, we notice that
α∈I2(λ1)
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0)xα
λ2j (2λ1 + λj)
β,γ∈I1(λ1)
α∈I2(λ1)
β+γ=α
k,ℓ=1
∂j∂k∂
βV (0)(g−1 (z
−))k∂j∂ℓ∂
γV (0)(g−1 (z
−))ℓ x
λ2j (2λ1 + λj)
α,β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
αV (0)
βV (0)
xα(g−1 (z
}(6.141)
42 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
From (6.124), (6.130), (6.131) (6.139), and (6.141), we finally obtain that
α∈I2(λ1)
c1,αx
γ∈I1\I1(2λ1)
α,β∈I2(λ1)
(2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))βxα
γ∈I1(2λ1)
α∈I2(λ1)
∂α+γV (0)
(z−))γxα +
γ∈I1(2λ1)
α,β∈I2(λ1)
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))βxα
α,β∈I2(λ1)
∂α+βV (0)
(g−1 (z
−))βxα
α,β∈I2(λ1)
(α+ β)!
γ,δ∈I2
γ+δ=α+β
(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
λj(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
(α + β)!
γ,δ∈I2(λ1)
γ+δ=α+β
λ2j(2λ1 + λj)
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
λ2j(2λ1 + λj)
αV (0)
βV (0)
xα(g−1 (z
−))β ,
(6.142)
or, more simply,
α∈I2(λ1)
c1,αx
α = −
γ∈I1(2λ1)
α∈I2(λ1)
∂α+γV (0)
(z−))γxα +
α,β∈I2(λ1)
(g−1 (z
γ∈I1\I1(2λ1)
(2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2
∂α+γV (0)∂β+γV (0)
γ∈I1(2λ1)
∂α+γV (0)∂β+γV (0)− ∂α+βV (0)
− (α+ β)!
γ,δ∈I2
γ+δ=α+β
γV (0)
δV (0)
λ2j (2λ1 + λj)
αV (0)∂j∂
βV (0)
.(6.143)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 43
7. Computations after the critical point
7.1. Stationary phase expansion in the outgoing region.
Now we compute the scattering amplitude starting from (4.19). First of all, we change the
cut-off function χ+ so that the support of the right hand side of the scalar product in (4.19)
is close to (0, 0).
ℓχ+ = 0
supp(∇χ+)
χ+ = 1
ℓeχ+ = 0
supp(∇eχ+)
eχ+ = 1
Figure 1. The support of χ+ and χ̃+ in T
Using Maslov’s theory, we construct a function v+ which coincides with a+(x, h)e
iψ+(x)/h
out of a small neighborhood of
∩ (B(0, R+ +1)×Rn) and such that v+ is a solution of
(P−E)v+ = 0 microlocally near
. Let χ̃+(x, ξ) ∈ C∞(T ∗Rn) such that χ̃+(x, ξ) = χ+(x)
out of a small enough neighborhood of
∩ (B(0, R++1)×Rn). In particular, (P −E)v+
is microlocally 0 near the support of χ+ − χ̃+. So, we have
〈u−, [χ+, P ]v+〉 =〈u−, [Op(χ̃+), P ]v+〉+ 〈u−, (χ+ −Op(χ̃+))(P − E)v+〉
− 〈(P − E)u−, (χ+ −Op(χ̃+))v+〉
=〈u−, [Op(χ̃+), P ]v+〉+O(h∞)− 〈g−eiψ−/h, (χ+ −Op(χ̃+))v+〉
=〈u−, [Op(χ̃+), P ]v+〉+O(h∞),(7.1)
since the microsupport of g−e
iψ−/h and χ+− χ̃+ are disjoint. Thus, the scattering amplitude
is given by
(7.2) A(ω, θ,E, h) = c(E)h−(n+1)/2〈u−, [Op(χ̃+), P ]v+〉+O(h∞).
Now we will prove that, modulo O(h∞), the only contribution to the scattering amplitude
in (7.2) comes from the values of the functions u− and v+ microlocally on the trajectories γ
and γ∞j . From (5.18), the fact that u− = O(h−C) and (P −E)u− = 0 microlocally out of the
microsupport of g−e
−iψ−/h, and the usual propagation of singularities theorem, we get
(7.3) MS(u−) ⊂ Λ−ω ∪ Λ+.
Moreover, we have
(7.4) MS(v+) ⊂ Λ+θ .
44 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Now, let f∞j (resp. f
ℓ ) be C
n) functions with support in a small enough neighborhood
of γ∞j (resp. γ
ℓ ∩MS(v+)) such that f∞j = 1 (resp. f
k = 1) in a neighborhood of γ
j (resp.
γ+ℓ ∩MS(v+)). In particular, we assume that all these functions have disjoint support. Since
u− and v+ have disjoint microsupport out of the support of the f
j and the f
ℓ , we have
A(ω, θ,E, h) =c(E)h−(n+1)/2
〈Op(f∞j )u−,Op(f∞j )[Op(χ̃+), P ]v+〉
+ c(E)h−(n+1)/2
〈Op(f+ℓ )u−,Op(f
ℓ )[Op(χ̃+), P ]v+〉+O(h
=Areg +Asing.(7.5)
Concerning the terms which contain f∞j , Areg, we are exactly in the same setting as in [30,
Section 4]. The computation there gives
(7.6) Areg =
j,m(ω, θ,E)h
iS∞j /h +O(h∞).
Now we compute Asing. Proceeding as in Section 5.2 for u−, one can show that v+ can be
written as
(7.7) v+(x) = a+(x, h)e
π/2eiψ+(x)/h,
microlocally near any ρ ∈ γ+
close enough to (0, 0). Here ν+
is the Maslov index of γ+
. The
phase ψ+ and the classical symbol a+ satisfy the usual eikonal and transport equations. In
particular, as in (5.28) and (5.33), we have
(7.8)
ℓ (t)) = −
|ξ+ℓ (u)|
2 − 2E01u>0 du = −
|ξ+ℓ (u)|
2 − V (x+ℓ (u)) −E0 sgn(u) du,
and a+(x, h) ∼
m a+,m(x)h
m with
(7.9) a+,0(x
ℓ (t)) = (2E0)
1/4(D+ℓ (t))
−1/2eitz,
where
(7.10) D+ℓ (t) =
∣∣ det
∂x+(t, z, θ, E0)
∂(t, z)
We can chose χ̃+ so that the microsupport of the symbol of Op(f
ℓ )[Op(χ̃+), P ] is contained
in a vicinity of such a point ρ ∈ γ+ℓ (see Figure 1). Then, microlocally near ρ, we have
(7.11) Op(f+
)[Op(χ̃+), P ]v+ = ã+(x, h)e
π/2eiψ+(x)/h,
(7.12) ã+(x, h) =
ã+,m(x)h
(7.13) ã+,0(x) = −i{χ̃+, p}(x,∇ψ+(x))a+,0(x).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 45
From [5, Section 5], the Lagrangian manifold
{(x,∇xϕk(t, x)); ∂tϕk(t, x) = 0},
coincides with Λ−ω . In particular, since MS(v+) ⊂ Λ+θ and since there is no curve γ∞(z∞j )
sufficiently closed to the critical point, the finite times in (6.5) give a contribution O(h∞)
to the scattering amplitude (4.19). In view of the equations (6.5), (6.12) and (7.11), the
principal contribution of Asing will come from the intersection of the manifolds Λ+θ and Λ+.
Recall that, from (A5), the manifolds Λ+θ and Λ+ intersect transversally along γ
In particular, to compute Asing, we can apply the method of stationary phase in the
directions that are transverse to γ+
. For each ℓ, after a linear and orthonormal change of
variables, we can assume that g+
) is collinear to the xℓℓℓ–direction, and that V (x) satisfies
(A2). We denote Hℓxℓℓℓ = {y = (y1, . . . , yn) ∈ R
n; yℓℓℓ = xℓℓℓ} the hyperplane orthogonal to
(0, . . . , 0, xℓℓℓ, 0, . . . , 0).
We shall compute Asing in the case where there is only one incoming curve γ−
and one
outgoing curve γ+
. In the general case, Asing is simply given by the sum over k and ℓ of such
contributions. Using (4.19), (6.5) and (7.11), we can write
Asing =c(E)h
−(n+1)/2
k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e
π/2dt dx
c(E)h−(n+1)/2√
y∈Hxℓℓℓ
k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e
π/2dt dy dxℓℓℓ.(7.14)
Let Φ(y) = ϕk(t, xℓℓℓ, y) − ψ+(xℓℓℓ, y) be the phase function in (7.14). From (6.10)–(6.13), we
can write
(7.15) Φ(y) = S−k + (ϕ+ − ψ+)(xℓℓℓ, y) + ψ̃(t, xℓℓℓ, y),
where ψ̃ = O(e−λ1t) is an expandible function. Since the manifolds Λ+
and Λ+ intersect
transversally along γ+
, the phase function y → (ϕ+−ψ+)(xℓℓℓ, y) has a non degenerate critical
point yℓ(xℓℓℓ) ∈ Hℓxℓℓℓ ∩ Πxγ
, and xℓℓℓ 7→ yℓ(xℓℓℓ) is C∞ for xℓℓℓ 6= 0. Then, from the implicit
function theorem, the function Φ has a unique critical point yℓ(t, xℓℓℓ) ∈ Hℓxℓℓℓ for t large enough
depending on xℓℓℓ. The function (t, xℓℓℓ) 7→ yℓ(t, xℓℓℓ) is expandible and we have
(7.16) yℓ(t, xℓℓℓ) = y
ℓ(xℓℓℓ)−Hess(ϕ+ − ψ+)−1
yℓ(xℓℓℓ)
yℓ(xℓℓℓ)
e−µ1t + Õ
e−µ2t
As a consequence, Φ
yℓ(t, xℓℓℓ)
is also expandible.
Since ϕ+ and ψ+ satisfy the same eikonal equation, we get (see (5.25))
(7.17) ∂t(ϕ+ − ψ+)(x+ℓ (t)) = |ξ
(t)|2 − |ξ+
(t)|2 = 0.
46 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Thus, (ϕ+ − ψ+)(yℓ(xℓℓℓ)) does not depend of xℓℓℓ and is equal to
(ϕ+ − ψ+)(yℓ(xℓℓℓ)) = lim
(ϕ+ − ψ+)(x+ℓ (t))
|ξ+ℓ (s)|
2 − 2E01s>0 ds
(s)|2 − V (x+
(s))− E0 sgn(s) ds
,(7.18)
where we have used (7.8). Therefore, the phase function Φ at the critical point yℓ(t, xℓℓℓ) is
equal to
yℓ(t, xℓℓℓ)
µm≤2λ1
t, yℓ(xℓℓℓ)
e−µmt
Hess(ϕ+ − ψ+)−1
yℓ(xℓℓℓ)
yℓ(xℓℓℓ)
· ∇ϕ1
yℓ(xℓℓℓ)
e−2µ1t + Õ(e−eµt),(7.19)
where µ̃ is the first of the µj’s such that µj > 2λ1.
Using the method of the stationary phase for the integration with respect to y ∈ Hℓxℓℓℓ in
(7.14), we get
(7.20) Asing = c(E)h
−(n+1)/2
(2πh)(n−1)/2
eiΦ(y
ℓ(t,xℓℓℓ))/hf ℓ(t, xℓℓℓ, h) dt dxℓℓℓ +O(h∞).
TheO(h∞) term follows from the fact that the error term stemming from the stationary phase
method can be integrated with respect to time t, since αk ∈ S0,2ReΣ(E), with ReΣ(E) > 0
(see the beginning of Section 6). The symbol f ℓ(t, xℓℓℓ, h) is a classical expandible function of
order S1,2ReΣ(E) in the sense of Definition 6.2:
(7.21) f ℓ(t, xℓℓℓ, h) ∼
f ℓm(t, xℓℓℓ, lnh)h
where the f ℓm are polynomials with respect to lnh and
(7.22) f ℓ0(t, xℓℓℓ, lnh) = α
t, yℓ(t, xℓℓℓ)
ã+,0
yℓ(t, xℓℓℓ)
π/2 e
i sgnΦ′′
|Hℓxℓℓℓ
(yℓ(t,xℓℓℓ))π/4
∣∣detΦ′′|Hℓxℓℓℓ
yℓ(t, xℓℓℓ)
)∣∣1/2
Using Proposition C.1, we compute the Hessian of Φ, and we get
yℓ(xℓℓℓ)
=diag(−λ1, . . . ,−λℓℓℓ−1, λℓℓℓ,−λℓℓℓ+1, . . . ,−λn) + o(1),
yℓ(xℓℓℓ)
=diag(λ1, . . . , λn) + o(1).
Then, for xℓℓℓ small enough and t large enough depending on xℓℓℓ, we have
∣∣detΦ′′|Hℓxℓℓℓ
yℓ(t, xℓℓℓ)
)∣∣1/2 =
j 6=ℓℓℓ
2λj + o(1),(7.23)
sgnΦ′′|Hℓxℓℓℓ
yℓ(t, xℓℓℓ)
= n− 1,(7.24)
as xℓℓℓ goes to 0.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 47
7.2. Behaviour of the phase function Φ.
Suppose that j ∈ N is such that j < ̂. From (6.40), we have
(7.25) ϕkj (x
(s0)) = e
−µj(s−s0)ϕkj (x
(s)).
Combining (6.41) with (6.109), we get
ϕkj (x
ℓ (s0)) =e
µjs0e−µjs
− 2µj〈g−j (z
k )|g
ℓ 〉)e
µjs +O(e2λ1s)
=− 2µj
g−j (z
∣∣g+j (z
eµjs0 .(7.26)
We suppose first that we are in the case (a) of assumption (A7). Then, (7.19) becomes
(7.27) Φ
yℓ(t, xℓℓℓ)
− 2µk
eµks(xℓℓℓ)e−µkt + Õ(e−µk+1t).
Here s(xℓℓℓ) is such that x
(s(xℓℓℓ)) = x
ℓ(xℓℓℓ) and the Õ(e−µk+1t) is in fact expandible, uniformly
with respect to xℓℓℓ when xℓℓℓ varies in a compact set avoiding 0.
Suppose now that we are in the case (b) of assumption (A7). Of course, from (7.26), we
have ϕj
yℓ(xℓℓℓ)
= 0 for all j < ̂. On the other hand, Corollary 6.8 and (6.111) imply
(7.28) ϕk
b,2(x
ℓ (s0)) = e
−2λ1(s−s0)ϕk
b,2(x
ℓ (s)).
Combining this with (6.126), we get
b,2(x
(s0)) =e
2λ1s0e−2λ1s
j∈I1(2λ1)
α,β∈I2(λ1)
∂α+1jV (0)
∂β+1jV (0)
g−1 (z
g+1 (z
e2λ1s
+O(e3λ1s)
j∈I1(2λ1)
α,β∈I2(λ1)
∂α+1jV (0)
∂β+1jV (0)
g−1 (z
g+1 (z
e2λ1s0 .(7.29)
In particular, (7.19) becomes, in that case,
yℓ(t, xℓℓℓ)
=S−k − S
j∈I1(2λ1)
α,β∈I2(λ1)
∂α+1jV (0)
∂β+1jV (0)
g−1 (z
g+1 (z
e2λ1s(xℓℓℓ)
× t2e−2λ1t +O(te−2λ1t)
=S−k + S
ℓ +M2(k, ℓ)t
2e−2λ1t +O(te−2λ1t).(7.30)
As in (7.27), the term O(te−2λ1t) is in fact expandible uniformly with respect to xℓℓℓ when xℓℓℓ
varies in a compact set avoiding 0.
Eventually, we suppose that we are in the case (c) of assumption (A7). Then we obtain
from (7.26) and (7.29) that ϕj
yℓ(xℓℓℓ)
= 0 for all j < ̂ and ϕb,2
yℓ(xℓℓℓ)
= 0. With the last
identity in mind, Equation (6.111) on ϕk
implies
(7.31) ϕk
b,1(x
ℓ (s0)) = e
−2λ1(s−s0)ϕk
b,1(x
ℓ (s)).
48 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
In order to compute ϕk
b,1(x
ℓ (s)), we put the expansion (2.15) for x
ℓ (s) (with Proposition 6.11
in mind) into (6.127). The third term in (6.127) will be, at least, O(e(µ2+µ1)s) = o(e2λ1s).
Thank to (6.91) and thanks to the fact that M2(k, ℓ) = 0, the first term in (6.127) will give
no contribution of order se2λ1s and will be of the form
(7.32) − 4λ1g−b,1(z
−) · x+ℓ (s) = −
α∈I2(λ1)
αV (0)
(g−1 (z
−))α(g+
(z+))je
2λ1s + Õ(eµb+1s)
It remains to study the contribution the second term in (6.127), as given in (6.143). As
previously, the first term of the third line in (6.143) will give a term of order o(e2λ1s). The
other terms will contribute to the order e2λ1s for
α∈I2(λ1)
αV (0)
(z−))j(g
+))α +
α,β∈I2(λ1)
(g−1 (z
(g+1 (z
− ∂α+βV (0) +
j∈I1\I1(2λ1)
λ2j (4λ
1 − λ2j )
∂α+γV (0)∂β+γV (0)
γ,δ∈I2(λ1)
γ+δ=α+β
(γ + δ)!
γ! δ!
γV (0)∂j∂
δV (0)
.(7.33)
Thus, combining (7.32) and (7.33), the discussion above leads to
b,1(x
ℓ (s0)) =e
2λ1s0e−2λ1s
M1(k, ℓ)e2λ1s + o(e2λ1s)
=M1(k, ℓ)e2λ1s0 .(7.34)
In particular, (7.19) becomes, in that case,
yℓ(t, xℓℓℓ)
=S−k + S
ℓ +M1(k, ℓ)e
2λ1s(xℓℓℓ)te−2λ1t +O(e−2λ1t).(7.35)
As above, the O(e−2λ1t) is expandible uniformly with respect to the variable xℓℓℓ when xℓℓℓ varies
in a compact set avoiding 0.
7.3. Integration with respect to time.
Now we perform the integration with respect to time t in (7.20). We follow the ideas of
[18, Section 5] and [5, Section 6]. Since yℓ(t, xℓℓℓ) is expandible (see (7.16)), and since Φ is C
outside of xℓℓℓ = 0, the symbol f
ℓ(t, xℓℓℓ, h) is expandible.
We compute only the contribution of the principal symbol (with respect to h) of f ℓ, since
the other terms can be treated the same way, and the remainder term will give a contribution
O(h∞) to the scattering amplitude. In other word, we compute
(7.36) Asing0 =
c(E)h−(n+1)/2√
(2πh)(n−1)/2h
eiΦ(y
ℓ(t,xℓℓℓ))/hf ℓ0(t, xℓℓℓ) dt dxℓℓℓ +O(h∞).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 49
First, we assume that we are in the case (a) of the assumption (A7). In that case, Ψ is
given by (7.27). For xℓℓℓ fixed in a compact set outside from 0, we set
yℓ(t, xℓℓℓ)
− (S−k + S
=− 2µk
eµks(xℓℓℓ)e−µkt +R(t, xℓℓℓ),(7.37)
and we perform the change of variable t→ τ in (7.36), and we assume for a moment that
(7.38)
Here R(t, xℓℓℓ) = Õ(e−µk+1t) is expandible. As in [18, Section 5] and [5, Section 6], we get
e−t ∼
− 2µk
(z−k )
(z+ℓ )
eµks(xℓℓℓ)
)−1/µkτ1/µk
τ bµj/µkbj(− ln τ, xℓℓℓ)
(7.39)
t ∼− 1
ln τ +
− 2µk
(z−k )
(z+ℓ )
eµks(xℓℓℓ)
τ bµj/µkbj(− ln τ, xℓℓℓ)(7.40)
τ bµj/µkbj(− ln τ, xℓℓℓ),(7.41)
where the bj ’s change from line to line. These expansions are valid in the following sense:
Definition 7.1. Let f(τ, y) be defined on ]0, ε[×U where U ⊂ Rm. We say that f = Ô(g(τ))
(resp. f = ô(g(τ))), where g(τ) is a non-negative function defined in ]0, ε[ if and only if for
all α ∈ N and β ∈ Nm,
(7.42) (τ∂τ )
α∂βy f(τ, y) = O(g(τ)),
(resp. o(g(τ))) for all (τ, y) ∈]0, ε[×U .
Thus, an expression like f ∼
j=1 τ
bµj/µkfj(− ln τ, xℓℓℓ), where fj(− ln τ, xℓℓℓ) is a polynomial
with respect to ln τ , like in (7.39)–(7.41), means that, for all J ∈ N,
(7.43) f(τ, x)−
τ bµj/µkfj(− ln τ, xℓℓℓ) = Ô(τ bµJ/µk).
We shall say that such symbols f are called expandible near 0.
Since f ℓ0(t, xℓℓℓ, h) is expandible (see Definition 6.1) with respect to t, this symbol is also
expandible near 0 with respect to τ in the previous sense. In particular, we get
(7.44) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ
τ (Σ(E)+cµj)/µk f̃ ℓ0,j(− ln τ, xℓℓℓ),
where the f̃ ℓ0,j’s are polynomials with respect to ln τ . The principal symbol f̃
0,0 is independent
on ln τ and we have
(7.45) f̃ ℓ0,0(xℓℓℓ) =
− 2µk
eµks(xℓℓℓ)
)−Σ(E)/µkf ℓ0,0(xℓℓℓ).
In that case, (7.36) becomes
(7.46) Asing0 =
c(E)h−1/2
(2π)1−n/2
∫∫ +∞
eiτ/hf̃ ℓ0(τ, xℓℓℓ)
dxℓℓℓ +O(h∞).
50 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1,
we can perform the integration with respect to t of each term in the right hand side of (7.44),
modulo a term O(h∞) (see (D.3)–(D.4) in Lemma D.1). Then, we get
(7.47) Asing0 =
c(E)h−1/2
(2π)1−n/2
f̂j(ln h)h
(Σ(E)+bµj )/µk ,
where f̂j(lnh) is a polynomial in respect to lnh. f̂0 does not depend on h and we have
(7.48) f̂0 = Γ(Σ(E)/µk)(−i)−Σ(E)/µk
f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ.
To finish the proof, it remains to perform the integration with respect to xℓℓℓ in (7.48). From
(7.22) and (7.45), it becomes
f̂0 =Γ(Σ(E)/µk)
(z−k )
(z+ℓ )
eµks(xℓℓℓ)
)−Σ(E)/µk
× α0,0(yℓ
xℓℓℓ)
ã+,0
yℓ(xℓℓℓ)
π/2 e
i sgnΦ′′
|Hℓxℓℓℓ
(yℓ(xℓℓℓ))π/4
∣∣ detΦ′′|Hℓxℓℓℓ
yℓ(xℓℓℓ)
)∣∣1/2
dxℓℓℓ.(7.49)
Now we make the change of variable xℓℓℓ 7→ s given by yℓ(xℓℓℓ) = x+ℓ (s) (then s(xℓℓℓ) = s). In
particular,
(7.50) dxℓℓℓ = ∂s(x
ℓ,ℓℓℓ(s))ds = λℓℓℓ|g
ℓℓℓ (z
ℓ )|e
λℓℓℓs(1 + o(1))ds,
as s→ −∞. In this setting, we get
(7.51) α0,0(x
(s)) = α0,0(0)(1 + o(1)),
as s→ −∞, where α0,0(0) is given in (6.8). We also have, from (7.9) and (7.13),
(7.52) ã+,0(x
ℓ (s)) = −i∂s
χ̃+(γ
ℓ (s))
(2E0)
1/4(D+ℓ )
−1/2eisz.
Then, putting (7.23), (7.24), (7.50), (7.51) and (7.52) in (7.49), we obtain
f̂0 =Γ(Σ(E)/µk)
(z−k )
(z+ℓ )
〉)−Σ(E)/µkα0,0(0)∂s
χ̃+(γ
ℓ (s))
i(n−1)π/4
j 6=ℓℓℓ
λℓℓℓ|g+ℓℓℓ (z
)|(2E0)1/4(D+ℓ )
−1/2eisze−Σ(E)seλℓℓℓs(1 + o(1)) ds
i(n+1)π/4
j 6=ℓℓℓ
)−1/2
λℓℓℓ|g+ℓℓℓ (z
)|Γ(Σ(E)/µk)
〉)−Σ(E)/µk
× e−iν
π/2α0,0(0)(2E0)
1/4(D+ℓ )
χ̃+(γ
ℓ (s))
(1 + o(1)) ds.(7.53)
Here the o(1) does not depend on χ̃+. Now, we choose a family of cut-off functions (χ̃
+)j∈N
such that the support of ∂t
goes to −∞ as j → +∞. We also assume that
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 51
ℓ (t))
is non negative (see Figure 1). Then
f̂0 =−
ei(n+1)π/4
j 6=ℓℓℓ
)−1/2
λℓℓℓΓ(Σ(E)/µk)e
π/2eiπ/4(2λ1)
3/2e−iν
× |g−1 (z
)| |g+
〉)−Σ(E)/µk(7.54)
(2E0)
1/2(D−
)−1/2 × (1 + o(1)).(7.55)
as j → +∞. Since f̂0 is also independent of χ̃+, we obtain Theorem 2.6 from (7.47) and
(7.48), in the case (a) and under the assumption (7.38). When 〈g−
(z−k )|g
(z+ℓ )〉 > 0, we set
τ as the opposite of the R.H.S. of (7.37), and we obtain the result along the same lines (see
Remark D.2).
Now we assume that we are in the case (b) of the assumption (A7). In that case, the
phase function Ψ is given by (7.30). For xℓℓℓ fixed in a compact set outside from 0, we set,
mimicking (7.37),
yℓ(t, xℓℓℓ)
− (S−k + S
=M2(k, ℓ)e2λ1s(xℓℓℓ)t2e−2λ1t +R(t, xℓℓℓ)(7.56)
where R(t, xℓℓℓ) = O(te−2λ1t) is expandible with respect to t. As above, we assume that
M2(k, ℓ) is positive (the other case can be studied the same way).
Following (7.39), we want to write s := e−t as a function of τ . Since t 7→ τ(t) is expandible
with respect to t, we have
(7.57) τ = M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ)),
where r(s, xℓℓℓ) = ô(1). In particular, ∂sτ > 0 for s positive small enough and then, for ε > 0
small enough, s 7→ τ(s) is invertible for 0 < s < ε. We denote s(τ) the inverse of this function.
We look for s(τ) of the form
(7.58) s(τ) = (2λ1)
M2(k, ℓ)e2λ1s(xℓℓℓ)
)1/2λ1 u(τ, xℓℓℓ)
(− ln τ)1/λ1
where u(τ, xℓℓℓ) has to be determined. Using (7.57), the equation on u is
τ =M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ))
=τu2λ1
(2λ1)
−2M2(k, ℓ)e2λ1s(xℓℓℓ)
+ 2λ1
− 2ln(− ln τ)
1 + r
(2λ1)
M2(k, ℓ)e2λ1s(xℓℓℓ)
)1/2λ1 u
(− ln τ)1/λ1
, xℓℓℓ
=τF (τ, u, xℓℓℓ),(7.59)
where F = u2λ1(1 + r̃(τ, u, xℓℓℓ)) and r̃ = ô(1) for u close to 1 (here (u, xℓℓℓ) are the variables y
in Definition 7.1). In other word, to find u, we have to solve F (t, u, xℓℓℓ) = 1.
First we remark that u 7→ F (τ, u, xℓℓℓ) is real-valued and continuous. Since, for δ > 0 and
τ small enough, F (τ, 1 − δ, xℓℓℓ) < 1 < (τ, 1 + δ, xℓℓℓ), there exists u ∈ [1 − δ, 1 + δ] such that
F (τ, 1 + δ, xℓℓℓ) = 1. Thank to the discussion before (7.58), the function s(τ) is of the form
(7.58) with u(τ, xℓℓℓ) ∈ [1− δ, 1 + δ], for τ small enough.
52 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
For τ > 0, the function F is C∞ and, since r̃ = ô(1), we have
(7.60) ∂u
F (τ, u, xℓℓℓ)− 1
(u(τ, xℓℓℓ)) = 2λ1u
2λ1−1(1 + oτ (1)) > λ1,
for τ small enough. The notation oτ (1) means a term which goes to 0 as τ goes to 0. Here
we have used the fact that u(τ, xℓℓℓ) is close to 1. In particular, the implicit function theorem
implies that u(τ, xℓℓℓ) is C
We write u = 1 + v(τ, xℓℓℓ) and we known that v ∈ C∞ and v = oτ (1). Differentiating the
equality
(7.61) 1 = F (τ, u(τ, xℓℓℓ), xℓℓℓ) =
u(τ, xℓℓℓ)
)2λ1(
1 + r̃(τ, u(τ, xℓℓℓ), xℓℓℓ)
one can show that v = ô(1). Thus we have
e−t =s(τ) = (2λ1)
M2(k, ℓ)e2λ1s(xℓℓℓ)
)1/2λ1 1 + r̂(τ, xℓℓℓ)
(− ln τ)1/λ1
,(7.62)
t =− ln τ
(1 + r̂(τ, xℓℓℓ)),(7.63)
+ r̂(τ, xℓℓℓ),(7.64)
where r̂(τ, xℓℓℓ) = ô(1) change from line to line.
Since f ℓ0(t, xℓℓℓ, h) is expandible with respect to t, we get, from (7.62)–(7.64),
(7.65) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ
= τΣ(E)/2λ1(− ln τ)−Σ(E)/λ1
f̃ ℓ0,0(xℓℓℓ) + r̂(τ, xℓℓℓ)
where r̂ = ô(1) and
(7.66) f̃ ℓ0,0(xℓℓℓ) = (2λ1)
Σ(E)/λ1−1
M2(k, ℓ)e2λ1s(xℓℓℓ)
)−Σ(E)/2λ1
f ℓ0,0(xℓℓℓ).
In that case, (7.36) becomes
(7.67) Asing0 =
c(E)h−1/2
(2π)1−n/2
∫∫ +∞
eiτ/hf̃ ℓ0(τ, xℓℓℓ)
dxℓℓℓ +O(h∞).
Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1,
we can perform the integration with respect to t in (7.67), modulo an error term given by
(D.3)–(D.4) in Lemma D.1. Then, we get
Asing0 =
c(E)h−1/2
(2π)1−n/2
)/hΓ(Σ(E)/2λ1)(−i)−Σ(E)/2λ1
× hΣ(E)/2λ1(− lnh)−Σ(E)/λ1
f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ + o(1)
,(7.68)
as h goes to 0. The rest of the proof follows that of (7.55).
At last, the proof of Theorem 2.6 in the case (c) can be obtained along the same lines, and
we omit it.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 53
Appendix A. Proof of Proposition 2.5
We prove that Λ+θ ∩Λ+ 6= ∅. From the assumption (A2), the Lagrangian manifold Λ+ can
be described, near (0, 0) ∈ T ∗(Rn), as
(A.1) Λ+ = {(x, ξ); x = ∇ϕ̃+(ξ)},
for |ξ| < 2ε, with ε > 0 small enough. For η ∈ Sn−1, let (x(t, η), ξ(t, η)) be the bicharacteristic
curve with initial condition (ϕ̃(εη), εη). We have
(A.2) Λ+ = {(x(t, η), ξ(t, η)); t ∈ R, η ∈ Sn−1} ∪ {(0, 0)}.
The function ξ(t, η) is continuous on R× Sn−1. From the classical scattering theory (see [13,
Section 1.3]), we know that this function ξ(t, η) converges uniformly to
(A.3) ξ(∞, η) := lim
ξ(t, η),
as t→ +∞ and ξ(∞, η) ∈
2E Sn−1.
Then, the function
(A.4) F (t, η) =
1−t , η)
|ξ( t
1−t , η)|
is well defined for 0 ≤ t ≤ 1 with the convention F (1, η) = ξ(∞, η)/
2E. Here we used that
|ξ(t, η)| 6= 0 for each t ∈ [0,+∞], η ∈ Sn−1. The previous properties of ξ(t, η) imply the
continuity of F (t, η) on [0, 1] × Sn−1.
From (A.2), to prove that Λ+θ ∩ Λ+ 6= ∅ for all θ ∈ Sn−1, it is enough (equivalent) to show
the surjectivity of η → F (1, η). But if η → F (1, η) is not onto, then ImF (1, ·) ⊂ Sn−1 \ {a
point}. And since Sn−1 \ {a point} is a contractible space, F (1, ·) is homotopic to a constant
(A.5) f : Sn−1 → Sn−1.
On the other hand, F : [0, 1]× Sn−1 −→ Sn−1 gives a homotopy between F (0, ·) = IdSn−1 and
F (1, ·). In particular, we have
(A.6) 1 = deg(F (0, ·)) = deg(F (1, ·)) = deg(f(·)) = 0,
which is impossible (see [16, Section 23] for more details).
Appendix B. A lower bound for the resolvent
Let χ ∈ C∞(]0,+∞[) be a non-decreasing function such that
(B.1) χ(x) =
x for 0 < x < 1
2 for 2 < x,
Let also ϕ ∈ C∞0 (R) an even function such that 0 ≤ ϕ ≤ 1, 1[−1,1] ≺ ϕ, and suppϕ ⊂ [−2, 2].
We set
(B.2) u(x) =
j/2hϕ
|xj |1/2
uj(x),
54 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where 0 < α < 2β will be fixed later on. The uj ’s are of course C
∞ functions, and we have
(B.3) (P − E0)u = −
∆u(x)−
x2ju(x) +O(x3u(x)).
Lemma B.1. For any h small enough, we have
hβn| lnh|n/2 . ‖u‖L2(Rn) . hβn| lnh|n/2,(B.4)
∥∥|x|3u(x)
L2(Rn)
. h3αhβn| lnh|n/2.(B.5)
Proof. First of all, the second estimate follow easily from the first one: we have
∥∥|x|3u(x)
∥∥2 =
|x|6|u(x)|2dx . h6α‖u‖2,
since u vanishes if |x| > 2hα. Thanks to the fact that u is a product of n functions of one
variable, it is enough to estimate
dt = 2
∫ 2hα
We have
dt ≤ I ≤ 2
∫ 2hα
dt+ 2
so that
dt ≤ I ≤ 2
∫ 2hα
dt+ 2
4 dt.
The first estimate follows from the fact that 2β − α > 0, once we have noticed that
∫ Ahα
dt = h2β
(2β − α)| ln h|+ α lnA
On the other hand, we have
∆u(x)−
x2ju(x) =
j 6=k
uj(xj)
u′′k(xk)−
x2kuk(xk)
From Lemma B.1, we get
∥∥(P − E0)u
∥∥ .hβ(n−1)| lnh|(n−1)/2 sup
1≤k≤n
∥∥h2u′′k(t) + λ
2uk(t)
∥∥+ h3αhβn| lnh|n/2
h−β | lnh|−1/2 sup
1≤k≤n
∥∥h2u′′k(t) + λ
2uk(t)
∥∥+ h3α
‖u‖.(B.6)
We also have
(B.7) h2u′′k(t) + λkt
2uk(t) = e
h2v′′h(t) + ihλk(2t∂t + 1)vh(t)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 55
where we have set vh(t) = ϕ
|t|1/2
. Notice that the right hand side of (B.7) is an even
function, so that we only have to consider t > 0. The point here, is that we have, for t > 0,
(B.8) (2t∂t + 1)
= − h


2 if 0 < t <
O(1) if h
< t < h2β ,
0 if h2β < t.
Therefore, we obtain
∥∥(2t∂t + 1)vh
∥∥2 =2
∫ 2hα
(2t∂t + 1)
|t|1/2
∫ 2hα
|t|1/2
∫ h2β
∫ 2hα
|t|1/2
dt . h2β .(B.9)
On the other hand, an easy computation gives, still for t > 0,
v′′h(t) =h
−2αϕ′′
4t5/2
.(B.10)
Computing the L2–norm of each of these terms as in Lemma B.1 and (B.9), we obtain
(B.11) ‖h2v′′h‖ . h2+β−2α + h2+β−2α + h2−3β + h2−3β ,
and, eventually, from (B.6), (B.7), (B.9) and (B.11),
∥∥(P − E0)u
h−β | lnh|−1/2
h1+β + h2+β−2α + h2−3β
+ h3α
Therefore we obtain Proposition 2.2 if we can find α > 0 and β > 0 such that
2− 2α > 1, 2− 4β > 1, 3α > 1 and 2β > α,
and one can check that α = 5/12 and β = 11/48 satisfies these four inequalities.
Appendix C. Lagrangian manifolds which are transverse to Λ±
Let Λ ⊂ p−1(E0) be a Lagrangian manifold such that Λ ∩ Λ− is transverse along a Hamil-
tonian curve γ(t) = (x(t), ξ(t)). Then, where exists a 6= 0 and ν ∈ {1, . . . , n} such that
(C.1) γ(t) = (a+O(e−εt))e−λν t,
as t→ +∞. The vector a is an eigenvector of
(C.2)
V ′′(0) 0
for the eigenvalue λν . Thus, up to a linear change of variable in R
n, we can always assume
that Πxa is collinear to the xν–direction. The goal of this section is to prove the following
geometric result.
56 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Proposition C.1. For t large enough, Λ projects nicely on Rnx near γ(t). In particular,
there exists ψ ∈ C∞(Rn) defined near Πxγ, unique up to a constant, such that Λ = Λψ :=
{(x,∇ψ(x)); x ∈ Rn}. Moreover, we have
(C.3) ψ′′(x(t)) =

. . .
. . .

+O(e−εt),
as t→ +∞.
Remark C.2. The same result hold in the outgoing region: If γ = Λ ∩ Λ+ is transverse,
Λ projects nicely on Rnx near γ(t), t → −∞. Then Λ = Λψ for some function ψ satisfying
ψ′′(x(t)) = diag(−λ1, . . . ,−λν−1, λν ,−λν+1, . . . ,−λn) +O(eεt).
Proof. We follow the proof of [18, Lemma 2.1]. There exist symplectic local coordinates (y, η)
centered at (0, 0) such that Λ− (resp. Λ+) is given by y = 0 (resp. η = 0) and
(ξj + λjxj) +O((x, ξ)2),(C.4)
(ξj − λjxj) +O((x, ξ)2).(C.5)
Then, p(x, ξ) = A(y, η)y · η with A0 := A(0, 0) = diag(λ1, . . . , λn).
(C.6)
A0 +O(e−λ1t) 0
O(e−λ1t) A0 +O(e−λ1t)
We denote by U(t, s) the linear operator such that U(t, s)δ solves (C.6) with U(s, s) = Id.
Since Λ∩Λ− = γ is transverse, there exists En−1(t0) ⊂ Tγ(t0)Λ, a vector space of dimension
n − 1 disjoint from Tγ(t0)Λ−. For convenience, we set En(t0) = En−1(t0) ⊕ Rv for some
v /∈ Tγ(t0)Λ + Tγ(t0)Λ−. Let E•(t) = U(t, t0)E•(t0). From [18, Lemma 2.1], there exists Bt =
O(e−λ1t) such that En(t) is given by δη = Btδx. Now, if δ ∈ En−1(t), we have σ(Hp, δ) = 0
since En−1(t)⊕ RHp = Tγ(t)Λ and Λ is a Lagrangian manifold. From (C.1), we have
(C.7) Hp(γ(t)) = γ̇(t) = −λν(ãeην +O(e−εt))e−λν t,
where eην is the basis vector corresponding to ην and then
(C.8) 0 = σ(eλν tHp, δ) = λν ãδyν +O(e−εt)|δ|.
It follows that δ ∈ En−1(t) if and only if (δyν , δη) = B̃tδy′ with B̃t = O(e−εt). Using Tγ(t)Λ =
En−1(t)⊕ RHp, we obtain that Tγ(t)Λ has a basis formed of vector fj(t) such that
fj =eyj +O(e−εt) for j 6= ν(C.9)
fν =eην +O(e−εt).(C.10)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 57
In the (x, ξ)-coordinates, Tγ(t)Λ has a basis formed of vector f̃j(t) of the form
f̃j =eξj + λjexj +O(e−εt) for j 6= ν(C.11)
f̃ν =eξν − λjexν +O(e−εt),(C.12)
and the lemma follows. �
Appendix D. Asymptotic behaviour of certain integrals
Lemma D.1. Let α ∈ C, Reα > 0, β ∈ R and χ ∈ C∞0 (]−∞, 1/2[) be such that χ = 1 near
0. As λ goes to +∞, we have
(D.1)
eiλttα(− ln t)βχ(t) dt
= Γ(α)(ln λ)β(−iλ)−α(1 + o(1)).
Moreover, if β ∈ N, we get
(D.2)
eiλttα(− ln t)βχ(t) dt
= (−iλ)−α
(j)(α)(−1)j
ln(−iλ)
+O(λ−∞).
Finally, if s(t) ∈ C∞(]0,+∞[) satisfies
(D.3) |∂jt s(t)| = o
tα−j(− ln t)β
for all j ∈ N and t→ 0, then
(D.4)
eiλts(t)χ(t)
(ln λ)βλ−α
Here (−iλ)−α = eiαπ/2λ−α and ln(−iλ) = lnλ− iπ/2.
Remark D.2. Notice that one obtains the behaviour of these quantities as λ → −∞ by
taking the complex conjugate in these expressions.
Proof. We begin with (D.2) and assume first that β = 0. Then, we can write
eiλttαχ(t)
= lim
ei(λ+iε)ttαχ(t)
= lim
I1(α, ε) + I2(α, ε)
,(D.5)
where
I1(α, ε) =
e−(ε−iλ)ttα
,(D.6)
I2(α, ε) =
ei(λ+iε)ttα(1− χ(t)) dt
·(D.7)
It is clear that
(D.8) I1(α, ε) = (ε− iλ)−αΓ(α),
where z−α is well defined on C\]−∞, 0] and real positive on ]0,+∞[. In particular
(D.9) lim
I1(α, ε) = (−iλ)−αΓ(α).
58 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Concerning, I2(α, ε), we remark that r(t, α) = t
α−1(1− χ(t)) is a symbol which satisfies
(D.10) |∂jt ∂kαr(t, α)| . 〈t〉Reα−1−j〈ln t〉k,
for all j, k ∈ N uniformly for t ∈ [0,+∞[ and α in a compact set of {Re z > 0}. Then, making
integration by parts in (D.7), we obtain
(D.11) I2(α, ε) =
(ε− iλ)j
e(iλ−ε)t∂jt r(t, α) dt,
for all j ∈ N. Now, if j is large enough (j > Reα), ∂jt r(t, α) is integrable in time uniformly
with respect to ε. In particular, for such j,
(D.12) lim
I2(α, ε) = e
ijπ/2λ−j
eiλt∂
t r(t, α) dt,
and then (see (D.10) or Cauchy’s formula)
∂kα lim
I2(α, ε) =e
ijπ/2λ−j
eiλt∂
αr(t, α) dt
=O(λ−∞),(D.13)
for all k ∈ N. Then we obtain (D.2) for β = 0. To obtain the result for β ∈ N, it is enough
to see that
eiλttα(ln t)βχ(t)
eiλttαχ(t)
(−iλ)−αΓ(α)
+ ∂βα lim
I2(α, ε)
=(−iλ)−α
(j)(α)
− ln(−iλ)
+O(λ−∞),(D.14)
from (D.13). Thus, (D.2) is proved.
Let u be a function C∞(]0,+∞[) be such that
(D.15) |∂jt u(t)| . tReα−j(− ln t)β,
near 0. Let ϕ ∈ C∞(R) such that ϕ = 1 for t < 1 and ϕ = 0 for t > 2. For δ > 0, we have
(D.16)
eiλtu(t)χ(t)
1−ϕ(t/δ)
= (−iλ)−N
eiλt∂Nt
u(t)χ(t)
1−ϕ(t/δ)
for all N .
If one of the derivatives falls on 1−ϕ(t/δ), the support of this contribution is inside [δ, 2δ].
Therefore, the corresponding term will be bounded by δReα−N−1(ln δ)β and will contribute
like δReα−N (− ln δ)β to the integral.
If ones of the derivatives falls on χ(t), the support of the integrand will be a compact set
outside of 0 and then this function will be O(1). The contribution to the integral of such
term will be like 1.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 59
If all the derivatives fall on u(t)t−1, we corresponding term will satisfies
eiλt∂Nt
u(t)t−1
1− ϕ(t/δ)
dt =O(1)
tReα−1−N (− ln t)β(1− χ(t))dt
.(− ln δ)βδReα−N ,(D.17)
for N large enough (N > Reα).
From this 3 cases, we deduce
(D.18)
eiλtu(t)χ(t)
1− ϕ(t/δ)
(− ln δ)βδα−Nλ−N
Taking δ = (ελ)−1, we get
(D.19)
eiλtu(t)χ(t)
1− ϕ(t/δ)
ε(lnλ)βλ−α
as λ→ +∞.
We now assume (D.3), and we want to prove (D.4). Since, for t small enough
(D.20) tReα−1(− ln t)β .
tReα(− ln t)β
we get
eiλts(t)χ(t)ϕ(t/δ)
∣∣∣ =oδ→0(1)
tReα−1(− ln t)βdt
=oδ→0(1)δ
Re α(− ln δ)β .(D.21)
Here oδ→0(1) stands for a term which goes to 0 as δ goes to 0. If δ = (ελ)
−1, we get
(D.22)
eiλts(t)χ(t)ϕ(t/δ)
∣∣∣ = oλ→+∞(1)λ−α(ln λ)β,
when λ → +∞ and ε fixed. Taking ε small enough in (D.19), and then λ large enough in
(D.22), we get (D.4).
We are left with (D.1). We need to compute
(D.23) I =
eiλttα(− ln t)βϕ(t/δ) dt
Performing the change of variable s = λt, we get
I =λ−α
∫ 2/ε
eissα(ln λ− ln s)βϕ(εs) ds
=(lnλ)βλ−α
∫ 2/ε
eissα(1− ln s/ lnλ)βϕ(εs) ds
.(D.24)
60 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
We remark that, in the previous equation, − ln s/ lnλ > − ln(2/ε)/ ln λ > −1/2 for λ large
enough. Using (1 + u)β = 1 +O(|u|+ |u|max(1,β)) for u > −1/2, we get
I =(lnλ)βλ−α
∫ 2/ε
eissαϕ(εs)
+ (lnλ)βλ−α
∫ 2/ε
sReαO
( | ln s|
( | ln s|
)max(1,β))
ϕ(εs)
=(lnλ)β
eiλttα(− ln t)βϕ(t/δ) dt
(lnλ)β−1λ−α
.(D.25)
Note that the Oε in (D.25) depends on ε.
Then, using (D.19), (D.25) and (D.19) again, we get
eiλttα(− ln t)βχ(t) dt
=I +O
ε(ln λ)βλ−α
=(ln λ)β
eiλttα(− ln t)βϕ(t/δ) dt
(lnλ)β−1λ−α
ε(lnλ)βλ−α
=(ln λ)β
eiλttα(− ln t)β dt
ε(lnλ)βλ−α
(lnλ)β−1λ−α
ε(ln λ)βλ−α
.(D.26)
Choosing ε small enough, then λ large enough, and using (D.2) with β = 0 to compute the
first term, we obtain
(D.27)
eiλttα(− ln t)βχ(t) dt
= Γ(α)(ln λ)β(−iλ)−α(1 + o(1)),
and this finishes the proof for (D.1). �
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http://arxiv.org/abs/math/0602069
62 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Ivana Alexandrova, Department of Mathematics, East Carolina University, Greenville, NC
27858, USA
E-mail address: alexandrovai@ecu.edu
Jean-François Bony, Institut de Mathématiques de Bordeaux, (UMR CNRS 5251), Université
de Bordeaux I, 33405 Talence, France
E-mail address: bony@math.u-bordeaux1.fr
Thierry Ramond, Mathématiques, Université Paris Sud, (UMR CNRS 8628), 91405 Orsay,
France
E-mail address: thierry.ramond@math.u-psud.fr
	1. Introduction
	2. Assumptions and main results
	3. Proof of the main resolvent estimate
	4. Representation of the Scattering Amplitude
	5. Computations before the critical point
	5.1.  Computation of u- in the incoming region
	5.2.  Computation of u- along -k
	6. Computation of u- at the critical point
	6.1.  Study of the transport equations for the phases
	6.2.  Taylor expansions of + and k1
	6.3.  Asymptotics near the critical point for the trajectories
	6.4.  Computation of the jk's
	7. Computations after the critical point
	7.1.  Stationary phase expansion in the outgoing region
	7.2.  Behaviour of the phase function 
	7.3.  Integration with respect to time
	Appendix A. Proof of Proposition 2.5
	Appendix B. A lower bound for the resolvent
	Appendix C. Lagrangian manifolds which are transverse to 
	Appendix D. Asymptotic behaviour of certain integrals
	References
