Himeji PDE Conference 2021

Resonances and complex absorbing potential
method for the Wigner-von Neumann type
Hamiltonian

Kentaro Kameoka
joint work with Shu Nakamura
to appear in Pure and Applied Analysis

1 Introduction
1.1 Overview
In this talk, we study the Wigner-von Neumann type Hamiltonian P ,which has a slowly decaying oscillatory potential.
A typical example iswhere a ∈ R.

sin 2xd2on L2 (R),P =− 2 +adxx

We deﬁne the set of resonances Res(P ) by a new complex distortion.
Moreover, we characterize resonances by the complex absorbing potential
method:lim σd (Pε ) = Res(P ),ε→0

where Pε = P − iεx2 with ε > 0.

1.2 Basic spectral properties of Schrödinger operators
We take V ∈ C(R; R) such that lim|x|→∞ V (x) = 0 and considerd2
P = − 2 + V (x) on L2 (R).
dx
Then we have σd (P ) ⊂ (−∞, 0) and σess (P ) = [0, ∞).
In this setting σd (P ) and σess (P ) basically correspond to bound statesand scattering states, respectively.
A naive reasoning is that even if there exist classical bound states at
positive energy, quantum states at positive energy will scatter to theinﬁnity by the tunneling eﬀect.
• There may be a positive eigenvalue; Wigner-von Neumann
Hamiltonian.
• Classical bound states at positive energy create “hidden complexeigenvalues”; resonances.

1.3 Wigner-von Neumann Hamiltonian
Wigner and von Neumann(1929) constructed a Hamiltoniand2
P = − 2 + VWvN (x) on L2 (R)dxwith VWvN (x) = −8 sinx2x + O(|x|−2 ) when |x| → ∞ such that 1 is aneigenvalue of P (or its spherically symmetric three dimensional
analogue).
The slow decay and oscillation are essential for the existence of positive
eigenvalue: Kato(1959), Simon(1967).
Hamiltonians with slowly decaying oscillatory potentials (Wigner-von
Neumann type Hamiltonian) have been investigated by many authors.

Problem 1: How about complex resonances?

1.4 What is resonance?
Resonances are the poles of meromorphic continuations of certain matrix
elements of the resolvent or cutoﬀ resolvent. They are analogous todiscrete eigenvalues, which are poles of the original resolvent.
Resonances represent quasi-steady states and they are also closely
related to scattering theory (cf. Breit-Wigner formula).
For instance, assume V (x) is analytic and decaying in{z = x + iy ∈ Cd ||x| > R0 , |y| < K|x|} for some R0 > 0 and K > 0(exterior dilation analyticity). Then for any 0 ̸≡ χ ∈ Cc∞ (Rd ), the cutoﬀ
resolvent χ(z − P )−1 χ has a meromorphic continuation from the upper
half plane to {z ∈ C| − 2 arctan K < arg z < π}. The poles are called
resonances of P and the set of resonances are denoted by Res(P ).

1.5 Complex absorbing potential methoddZworski(2018) proved that for P = −∆ + V with V ∈ L∞(R) andcomp
Pε = P − iεx2 with ε > 0,lim σd (Pε ) = Res(P )ε→0in z ∈ C − π4 < arg z ≤ 0 .

The proof employed the dilation analytic method.
Complex absorbing potential method is used in physical chemistry.
In the Fourier space, this is analogous to the viscosity limit for 1st orderPDEs.
Zworski proposed a problem of ﬁnding V such that limε→0 σd (Pε ) is notdiscrete and took V = sinx2x as a candidate.

Problem 2: Study limε→0 σd (Pε ) for the Wigner-von Neumann type
Hamiltonian P .

2 Results
2.1 Assumption
Assumption 1. The potential V (x) has the following form:
JXV (x) =sj (x)Wj (x),j=1where J ∈ N, sj ∈ C(R; R) are periodic functions with period π whose
Fourier series converge absolutely, and Wj ∈ C ∞ (R; R) have analytic
continuations to the region {z = x + iy ∈ C | |x| > R0 , |y| < K|x|} forsome R0 > 0 and K > 0 with the bound |Wj (z)| ≤ C|z|−µ for someµ > 0 in this region.
• A typical example is V (x) = a sinx2x for a ∈ R.
• By taking sj (x) = 1, dilation analytic potentials are included.
• We work in one space dimension. How about the spherically symmetrichigher dimensional case?
• Unfortunately, the original Wigner-von Neumann potential
VWvN (x) = (1+g(x)2 )−2 (−32 sin x)(g(x)3 cos x− 3g(x)2 sin3 x + g(x) cos x + sin3 x),
where g(x) = 2x − sin 2x, does not seem to satisfy the analyticity
condition of Assumption 1.

Figure 1 The domain of analyticity

2.2 Result 1
We set R+ (z) = (z − P )−1 for Imz > 0. We write the set of thresholds
by T = {n2 | n ∈ N ∪ {0}}.
Theorem 1. Under Assumption 1, there exists a complex neighborhoodΩ ⊂ C of [0, ∞) \ T such that the following holds: For any
f, g ∈ L2comp (R), the matrix element (f, R+ (z)g) has a meromorphic
continuation to Ω.

Figure 2 The domain Ω in theorems

2.3 Deﬁnition of resonances
We deﬁne resonances of P using Theorem 1.
Deﬁnition 1. A complex number z ∈ Ω is called a resonance if z is apole of (f, R+ (z)g) for some f, g ∈ L2comp (R). The multiplicity mz isdeﬁned as the maximal number m such that there exist
f1 , . . . , fm , g1 , . . . , gm ∈ L2comp (R) withmH
1
det 2πi C(z) (fi , R+ (ζ)gj )dζ i,j=1 ̸= 0, where C(z) is a small circle
around z. The set of resonances is denoted by Res(P ).
Remark 2.1. Res(P ) is discrete in Ω and mz < ∞ for any z ∈ Ω.
• If z is not a resonance, we set mz = 0.

2.4 Result 2We recall Pε = P − iεx2 , where ε > 0. We easily see that Pε , ε > 0, has
purely discrete spectrum on L2 (R).
Theorem 2. Under Assumption 1, there exists a complex neighborhoodΩ ⊂ C of [0, ∞) \ T such that limε→0 σd (Pε ) = Res(P ) in Ω including
multiplicities. In particular, limε→0 σd (Pε ) is discrete in Ω. Moreprecisely, for any z ∈ Ω there exists ρ0 > 0 such that for any 0 < ρ < ρ0there exists ε0 > 0 such that for any 0 < ε < ε0 ,#σd (Pε ) ∩ B(z, ρ) = mz ,where B(z, ρ) = {w ∈ C | |w − z| ≤ ρ}.

• In particular, this disproves the conjecture by M. Zworski outside thethresholds.

3 Idea of the proofs
3.1 Review of complex distortion
Global dilation analyticity: Aguilar-Combes(1971).
We set A = 12 (x · D + D · x) with D = 1i ∂. Then the unitary groupUθ = eiθA generated by A is
1iθA′Uθ f = e f (x) = (det Φθ (x)) 2 f (Φθ (x)) = edθ/2 f (eθ x)
for f ∈ L2 (Rd ). Here the dilation Φθ (x) = eθ x is the ﬂow of the vector
ﬁeld v(x) = x.
Consider the Schrödinger operator P = −∆ + V and deﬁnePθ = Uθ P Uθ−1 for θ ∈ R. Note that Uθ (−∆)Uθ−1 = e−2θ (−∆) has ananalytic continuation with respect to θ.
Assume that Vθ = Uθ V Uθ−1 also has an analytic continuation withrespect to θ (and −∆-compact). Then Pθ is well-deﬁned for complex θand σess (Pθ ) = e−2iImθ [0, ∞).

The simple formula(f, R+ (z)g) = (Uθ f, (z − Pθ )−1 Uθ g)
is valid for complex θ if f, g are analytic vectors for A. We considerImθ > 0. This implies that (f, R+ (z)g) has a meromorphic continuation
and resonances are deﬁned in {z ∈ C| − 2Imθ < arg z ≤ 0}. Moreover,resonances coincide with discrete eigenvalues of Pθ .

Hunziker-type distortion: Hunziker(1986).
Take a vector ﬁeld v on Rd such that v(x) = x for |x| ≫ 1 and v(x) = 0for |x| ≤ R0 + 1, and set Φθ (x) = x + θv(x) rather than the ﬂow of v.
1′2Set Uθ f (x) = (det Φθ (x)) f (Φθ (x)) and Pθ = Uθ P Uθ−1 .
Then the resonance theory can be constructed under the assumption that
V (x) is analytic and decaying in {x ∈ Cd ||Rex| > R0 , |Imx| < K|Rex|}
for some R0 > 0 and K > 0. Then Pθ has an analytic continuation withrespect to θ and resonances coincide with its eigenvalues.

There are other distortions;

Momentum distortion: Sigal(1984), Cycon(1989), Nakamura(1990).
Complex distortion on the Fourier space for exponentially decaying ordilation analytic potentials.

Microlocal distortion: Helﬀer-Sjöstrand(1986).
Based on microlocal and geometric viewpoints. Powerful but diﬃcult.

Geometric complex scaling: Sjöstrand-Zworski(1991).
Distortion on the conﬁguration space. Based on more geometricviewpoint than Hunziker. This enables us to study resonances in a larger
angle in the complex plane.

3.2 Relation with Mourre inequality
General idea Let P and A be self adjoint operators.
Formally speaking, analyticity of Pθ = eiθA P e−iθA would mean that∞X
1ad(iθA)j Pj!
j=0
converges for complex θ. If θ = iδ with δ > 0, the leading part of theimaginary part of Pθ is −δ[P, iA]. Thus, to deﬁne the resonances, it will
be enough to ﬁnd A such that
1. [P, iA] is positive for some energy interval I of P , that is the Mourre
inequality with a compact remainder K holds;
EI (P )[P, iA]EI (P ) ≥ cEI (P ) + K
in the quadratic form sense.
2. The repeated commutators ad(iA)j P are well-behaved for j ≥ 0.
This does not hold for our P with A = 12 (x · D + D · x) since Dj V (x)does not decay faster than V (x).

3.3 Mourre estimate with diﬀerence conjugate operator
Nakamura(2015) showed that the Mourre theory can be constructedusing a diﬀerence conjugate operator
11′′′′A = (x · D + D · x), D u(x) =(u(x + π) − u(x − π))22πid2π
under the assumption that x∆ V is − dx2 -compact and x2 ∆π ∆π V isd2− dx2 -bounded, where ∆π u(x) = π1 (u(x + π) − u(x)).
In the Fourier space, A′ is a diﬀerential operator
1′f
A =(i∂ξ ) · sin(πξ) + sin(πξ) · (i∂ξ ) .
2πThus22 f′[ξ , iA ] = sin(πξ)ξ
πis positive or negative when |ξ| ̸∈ Z≥0 . Thus the Mourre inequality holds
for ±A′ in [0, ∞) \ T . (In higher dimensional case, we can treat low
energy regime (0, 1).)Moreover, ad(iA′ )j V is bounded any j ≥ 0 if V = a sinx2x .

3.4 Periodic Fourier complex distortion
The above considerations lead us to consider eby the ﬂow of sin(πξ).

f′iθ A, which is the pullbackf′

In place of eiθA , we take Hunziker-type distortion in the Fourier space1′2Φθ (ξ) = ξ + θ sin(πξ), Uθ f (ξ) = Φθ (ξ) f (Φθ (ξ)),
where θ ∈ (−π −1 , π −1 ).
In the Fourier space, P has the form Pe = ξ 2 + Ve , where
Ve = (2π)−1/2 V̂ ∗ is a convolution
operator and V̂ is the FourierRtransform V̂ (ξ) = (2π)−1/2 V (x)e−ixξ dx.
Then we deﬁne our distorted operatorPeθ := Uθ PeUθ−1 = (ξ + θ sin(πξ))2 + Veθ ,This is our periodic Fourier complex distortion.

Veθ = Uθ Ve Uθ−1 .

3.5 Analyticity of Veθ
Lemma 1. Under Assumption 1, Veθ is analytic with respect to θ andξ 2 -compact for θ in some complex neighborhood of{iδ | −Kπ −1 < δ < Kπ −1 }, where K is the constant in Assumption 1.
Moreover, for any γ > 0 there is a decomposition Veθ = Veθ,1 + Veθ,2 suchthat ∥Veθ,1 ∥H −N →H N < ∞ for any N > 0 and ∥Veθ,2 ∥L2 →L2 < γ.
This is the most technical part of our arguments and we omit it. But forthe model case V (x) = a sinx2x , the proof is much simpler. In this case,′ 12 e′ 12e
one can show that Vθ f (ξ) = (Φθ ) V (Φθ ) f (ξ) by simple computations
using the special form of Ve = a2 χ[−2,2] ∗. Then the Lemma 1 for thiscase follows by an easy argument.

3.6 Essential spectrum of deformed operator
Lemma 1 implies that Peθ is analytic withthe essential
 respect to θ andspectrum of Peθ is given by σess (Peθ ) = (ξ + θ sin(πξ))2 ξ ∈ R .
For the energy interval ((n − 1)2 , n2 ), we take θ = (−1)n iδ = ±iδ. We
set δ0 = min{π −1 , Kπ −1 } and consider 0 < δ < δ0 . We set Ωn,δ as in
Figure 4.

Figure 3 σess (Peθ ) for θ = 0.2i.

Figure 4 The region Ωn,δ .

3.7 Deﬁnition of resonances
Then Theorem 1 is true for Ω =Sn∈N Ωn,δ0 .

Basically, the proof of Theorem 1 follows the standard arguments ofresonance theory once we prove the analyticity of Peθ .
Here are some remarks:• For the set of analytic vectors, we take A = L2comp (R). Since theFourier transform of f ∈ A is analytic, Uθ fˆ is analytic with respect to θ.
fθ including• Resonances coincide with the discrete eigenvalues of P
multiplicities using the formulafθ )−1 Uθ ĝ).
(f, R+ (z)g) = (Uθ fˆ, (z − P
In particular, Res(P ) is discrete and mz < ∞ for any z ∈ Ω.

3.8 Viscosity limit
The proof follows the strategy of Zworski(2018) replacing complex
dilation by our periodic Fourier complex distortion.
In the Fourier space, Pε has the following form:
Peε = ξ 2 + Ve + iε∂ξ2 .
We set Peε,θ = Uθ Peε Uθ−1 . Then σd (Peε,θ ) = σd (Peε ) including
multiplicities by the analyticity with respect to θ.
Thus it is enough to show limε→0 σd (Peε,θ ) = σd (Peθ ). We prove this byGohberg-Sigal theory and their operator valued Rouché’s theorem.
We ﬁrst consider the distorted free Hamiltonian (Peε,θ with V = 0)e ε,θ = (ξ + θ sin(πξ))2 − iεDξ (1 + πθ cos(πξ))−2 Dξ − iεrθ (ξ), ε ≥ 0.
QWe employ the semiclassical microlocal analysis in the Fourier space with√the semiclassical parameter h = ε to obtain a resolvent estimatee ε,θ − w)−1 ∥L2 →L2 ≤ C∥(Qfor small ε > 0. The semiclassical limit corresponds to the viscosity limit.
By a perturbation argument and operator valued Rouché’s theorem, itsuﬃces to show
e 0,θ − w)−1 − Veθ (Qe ε,θ − w)−1 ∥L2 →L2 = 0.
lim ∥Veθ (Q(1)ε→0By the second part of Lemma 1, we can replace Veθ by a smoothing Veθ,1 .
By the resolvent equation, we also learne 0,θ − w)−1 − Veθ,1 (Qe ε,θ − w)−1Veθ,1 (Qe 0,θ − w)−1 (Dξ (1 + πθ cos(πξ))−2 Dξ + rθ (ξ))(Qe ε,θ − w)−1 .
= −iεVeθ,1 (QSince Veθ,1 : H −2 → L2 ,e 0,θ − w)−1 − Veθ,1 (Qe ε,θ − w)−1 ∥L2 →L2 ≤ Cε,∥Veθ,1 (QS
which completes the proof. We can take Ω = n∈N Ωn,δ1 with√δ1 = min{( 2 − 1)π −1 , Kπ −1 }.

4 Summary• We deﬁned the resonances for the Wigner-von Neumann type
Hamiltonian by introducing periodic complex distortion in the Fourierspace.
• We also characterized resonances as limit points of discrete
eigenvalues in the viscosity-type limit.

Problems• Relation with quasi-steady states.
• Relation with scattering theory.
• Spherically symmetric higher dimensional case.
• Resonances for the original Wigner-von Neumann Hamiltonian.
• Structure of resolvent near threshold.

Thank you for listening!