数理解析研究所講究録 1255 巻 2002 年 55-75

55

HIGH ENERGY AND SMOOTHNESS ASYMPTOTIC EXPANSION OF
THE SCATTERING AMPLITUDE(MAIN IDEAS OF THE APPROACH)
D.Yafaev
Department of Mathematics, University Rennes-l, Campus Beaulieu, 35042,
Rennes, France ( e-mail : yafaev@univ-rennes1.fr )Abstract
We find an explicit expression for the kernel of the scattering matrix for the
Schr\"odinger operator containing at high energies all terms of power order. It turns
out that the same expression gives acomplete description of the diagonal singularities of the kernel in the angular variables. The formula obtained is in some sense
universal since it applies both to short- and long-range electric as well as magnetic
potentials.

1.

INTRODUCTION

for the:
1. High energy asymptotics of the scattering matrix, $d\geq 2$ , with areal short-range
Schrodinger operator $H=-\Delta+V$ in the spacepotential (bounded and satisfying the condition $V(x)=O(|x|^{-\rho})$ , $\rho>1$ , as $|x|arrow\infty$ ) is,
given by the Born approximation. To describe it, let us introduce the operator$S(\lambda)$$L_{2}(\mathrm{S}^{d-1})arrow L_{2}(\mathrm{S}^{d-1})$$??=L_{2}(\mathrm{R}^{d})$$\Gamma_{0}(\lambda)$$(\Gamma_{0}(\lambda)f)(\omega)=2^{-1/2}k^{(d-2)/2}\hat{f}(k\omega)$,$k=\lambda^{1/2}\in \mathrm{R}_{\dagger}=(0, \infty)$,$\omega$$\in \mathrm{S}^{d-1}$

,

(1.1)

of the restriction (up to the numerical factor) of the Fourier transform of afunction to
the sphere of radius . Set $R_{0}(z)=(-\Delta-z)^{-1}$ , $R(z)=(H-z)^{-1}$ . By the Sobolev: $H$ $arrow L_{2}(@^{d-1})$
theorem and the limiting absorption principle the operators{ are correctly defined as bounded operators for any
: $H$
and
, respectively. Therefore it isand$r>1/2$ and their norms are estimated byeasy to deduce (see, e.g., [14, 24]) from the usual stationary representation$f$$\hat{f}$$k$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\Gamma_{0}(\lambda)\langle x\rangle^{-r}$$(x)^{-r}R(\lambda+i0)\langle x\rangle^{-r}$$arrow\gamma$$\lambda^{-1/4}$$\lambda^{-1/2}$(1.2)$S(\lambda)=I-2\pi i\Gamma_{0}(\lambda)(V-VR(\lambda+i0)V)\Gamma_{0}^{*}(\lambda)$

for the scattering matrix (SM) and the resolvent identity that
$S( \lambda)=I-2\pi i\sum_{n=0}^{N}(-1)^{n}\Gamma_{0}(\lambda)V(R_{0}(\lambda+i0)V)^{n}\Gamma_{0}^{*}(\lambda)+\sigma_{N}(\lambda)$

where $||\sigma_{N}(\lambda)||=O(\lambda^{-(N+2)/2})$ as
Schatten -von Neumann classes. Moreover, the operators
and $\alpha(N)arrow 0$ as $Narrow\infty$ .

$\lambdaarrow\infty$$\mathfrak{S}_{\alpha(N)}$$\sigma_{N}$

,

(1.3)

belong to suitable

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Nevertheless the Born expansion (1.3) has at least three drawbacks. First, the structure of theterm is extremely complicated already for relatively small . Second, (1.3)
definitely fails for long-range potentials, and, finally, it fails as
for aperturbationof theby first order differential operators even with short-range coefficients$n^{\mathrm{t}\mathrm{h}}$$n$$\lambdaarrow\infty$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}-\Delta$(magnetic potentials).

2. In the particular case when $A=0$ and belongs to the Schwartz class aconvenient form of the high-energy expansion of the kernel of SM (called often the scattering
amplitude) was obtained in [3] (see also the earlier paper [7]). The method of [3] relies onapreliminary study of the scattering solutions of the Schr\"odinger equation defined, forexample, by the formula$V$$\psi_{\pm}(\xi)=u_{0}(\xi)-R(|\xi|^{2}\mp i0)Vu_{0}(\xi)$,$u_{0}(x, \xi)=\exp(i\langle x, \xi\rangle)$,$\xi=\hat{\xi}|\xi|\in \mathrm{R}^{d}$

It is shown in [3] that (at least on all compact sets of ) the function
asymptotic expansion
where$\psi_{\pm}(x,\xi)$$x$.

has the$\psi_{\pm}(x,\xi)=e^{*(x,\xi\}}.\mathrm{b}_{\pm}(x, \xi)$$\mathrm{b}_{\pm}(x,\xi)=\mathrm{b}_{\pm}^{(N)}(x, \xi)=\sum_{n=0}^{N}(2i|\xi|)^{-n}b_{n}^{(\pm)}(x, \xi)$,$\mathrm{b}(x,\xi)=1$,

N$arrow\infty$

.

(1.4)

The function
is determined by the transport equation (see subs. 2.3 below), andthe coefficientsare quite explicit. Therefore it is easy to deducefrom (1.2) that, for any $N$ , the kernel of SM admits the asymptotic expansion$\mathrm{b}_{\pm}(x,\xi)$$b_{n}^{(\pm)}(x, \xi)=b_{n}^{(\pm)}(x,\hat{\xi})$$s(\omega,\omega’;\lambda)=\delta(\omega,\omega’)-\pi i(2\pi)^{-d}k^{d-2}$$\cross\sum_{n=0}^{N}(2ik)^{-n}\int_{\mathrm{R}^{d}}e^{\dot{|}k(\omega’-\{v,x)}V(x)b_{n}^{(-)}(x,\omega’)dx+O(k^{d-3-N})$

,

(1.5)

where
is of course the Dirac-function on the unit sphere. We emphasize that thefunctions
are growing as $|x|arrow\infty$ in the direction of and the rate of growthincreases as increases. Thus, expansion (1.5) loses the sense (for sufficiently large $N$ ) if
$V(x)$ decreases only as some power of.
$\delta(\cdot)$$b_{n}^{(-)}(x,\omega’)$$\omega’$$n$$|x|^{-1}$The generalization of the results of [3] to the case of short-range potentials sat isfying the condition
for some $\rho_{v}>1$ was suggested in [21] where
the asymptotics of the scattering amplitude was also deduced from that of the scattering solutions. We note finally the paper [4] where the leading term of the high-energy
asymptotics of the scattering amplitude was found for short-range magnetic potentials.
$V$$\mathrm{d}\mathrm{a}\mathrm{V}(\mathrm{x})=O(|x|^{-\rho_{v}-|\alpha|})$3. In the present paper we suggest anew method which allows us to find an explicit
function
which describes with arbitrary accuracy the kernelof theat high energies (as).both for short- and long-range electric and magnetic potentials. It turns out that the same function
gives also all diagonal
singularities of the kernelin the angular variables. We emphasizethat our approach allows us to avoid astudy of solutions of the Schr\"odinger equation.
We consider the Schrodinger operator$s_{0}(\omega,\omega’;\lambda)$$\mathrm{S}\mathrm{M}S(\lambda)$$s(\omega,\omega’;\lambda)$$\lambdaarrow\infty$$s_{0}(\omega,\omega’;\lambda)$$s(\omega,\omega’;\lambda)$

H $=(i\nabla+A(x))^{2}+\mathrm{V}(\mathrm{x})$$\omega,\omega’\in \mathrm{S}^{d-1}$

(1.3)

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in the space H with electric $V(x)$ and magnetic $A(x)=(A_{1}(x),$satisfying the assumptions$A_{d}(x))$\ldots ,$|\partial^{\alpha}V(x)|$$|\partial^{\alpha}A(x)|$$\leq$$C_{\alpha}(1+|x|)^{-\rho_{v}-|\alpha|},$$\leq$$C_{\alpha}(1+|x|)^{-\rho_{a}-|\alpha|}$$\rho_{a}>0\rho_{v}>0,$,,

potentials

(1.7)

$\}$

for all multi-indices . We suppose that potentials are real, that is $V(x)=\overline{V(x)}$ and, and, $j=1$ ,, . Set$\alpha$$A_{j}(x)=\overline{A_{j}(x)}$$\ldots$$d$$\rho=\min\{\rho_{v}, \rho_{a}\}$I4(x) $=V(x)+|A(x)|^{2}$ ,$V_{1}(x)=V_{0}(x)+i\mathrm{d}\mathrm{i}\mathrm{v}A(x)$Then$H=-\Delta+2i\langle A(x), \nabla\rangle+V_{1}(x)$.

.

We emphasize that the cases $\rho>1$ (short-range potentials) and $\rho\in(0, 1]$ (long-range
potentials) are treated in almost the same way.
Let us formulate our main result. The answer is given in terms of approximate solutionsof the Schr\"odinger equation-A (x) $\xi)+2i\langle A(x), \nabla\rangle\psi(x, \xi)+V_{1}(x)\psi(x, \xi)=|\xi|^{2}\psi(x,\xi)$ .

(1.8)

To be more precise, we denote by $u_{\pm}(x, \xi)=u_{\pm}^{(N)}(x, \xi)$ explicit functions (see Section 2,
for their construction)(1.9)$u_{\pm}(x, \xi)=e^{i\mathrm{e}_{\pm(x,\xi)}}\mathrm{b}_{\pm}(x, \xi)$such that$(-\Delta+2i\langle A(x), \nabla\rangle+V_{1}(x)-|\xi|^{2})u_{\pm}(x, \xi)=e^{i\mathrm{e}_{\pm(x,\xi)}}r_{\pm}(x, \xi)=:q\pm(x, \xi)$(1.10)

as
as $|x|arrow\infty$ andand $r_{\pm}(x, \xi)=r_{\pm}^{(N)}(x, \xi)$ tends to zero faster than$Narrow\infty$ off any conical neighborhood of$q=q(N)arrow\infty$$p=p(N)arrow\infty$
as
where
and
if $A(x)=0$ and $V(x)$ is a
. Note that the phasethe direction
short-range function and $_{\pm}(x, \xi)$ satisfies the eikonal equation in the general case. The
is obtained as asolution of the corresponding transport equation.
function$|x|^{-p}$$\hat{x}=\mp\hat{\xi}$$|\xi|^{-q}$$|\xi|arrow\infty$$_{\pm}(x, \xi)=\langle x, \xi\rangle$$\mathrm{b}_{\pm}(x, \xi)$is a, the kernel
As is well known (see [1]), off the diagonal
.
A
of
as
power
any
faster
than
,
to
zero
of
where
it
tends
functionin aneighborhood of the diagonal
Thus, it suffices to describe the structure ofandbe the plane orthogonal to. Let
be an arbitrary point,.
Set
be determined by the condition$\omega=\omega’$$\omega$$C^{\infty}-$$s(\omega, \omega’;\lambda)$$\lambda^{-1}$$\omega’\in \mathrm{S}^{d-1}$$arrow\infty$$s(\omega, \omega’;\lambda)$$\omega=\omega’$$\omega_{0}\in \mathrm{s}^{d-1}$$\Pi_{\omega_{0}}$$\omega_{0}$$\pm\langle\omega, \omega_{0}\rangle>\delta>0$$\Omega_{\pm}(\omega_{0}, \delta)\subset \mathrm{S}^{d-1}$$x=\omega_{0}z+y$,$y\in\Pi_{\omega_{0}}$,(1.11)

and$s_{0}(\omega, \omega’;\lambda)=s_{0}^{(N)}(\omega, \omega’;\lambda)=\mp\pi ik^{d-2}(2\pi)^{-d}$$\cross(\int_{\mathrm{n}_{\omega_{0}}}(\overline{u_{+}(y,k\omega)}(\partial_{z}u_{-})(y, k\omega’)-u_{-}(y, k\omega’)\overline{(\partial_{z}u_{+})(y,k\omega)})dy$$-2i \int_{\mathrm{n}_{\omega_{0}}}\langle A(y),\omega_{0}\rangle\overline{u_{+}(y,k\omega)}u_{-}(y, k\omega’)dy)$(1.10)

58
forkernel$\omega,\omega’\in\Omega_{\pm}=\Omega_{\pm}(\omega_{0}, \delta)$. Then, for any ,
$p$

$q$

and sufficiently large $N=N(p, q)$ , the(1.13)$\tilde{s}^{(N)}(\omega,\omega’;\lambda)=s(\omega,\omega’\cdot\lambda)-s_{0}^{(N)}(\omega,\omega’;\lambda)$

belongs to the class, and its
wheremore is.
as A
Thus, all singularities ofboth for high energies and in smoothness are describedby the explicit formula (1.12). Let
be integral operator with kernel. Inview of representation (1.9), formula (1.12) shows that we actually consider the singular
partof the SM as aFourier integral or, more precisely, apseud0-differential operator(PDO) acting on the unit sphere and determined by its amplitude.
By our construction of functions (1.9), $u_{+}(x,\xi)=\overline{u_{-}(x,-\xi)}$ if $A(x)=0$ . Therefore
in the case $A=0$ the singular partsatisfies the same symmetry relation (thetime reversal invariance)$C^{p}$$\Omega=\Omega_{+}\cup\Omega_{-}$$C^{p}(\Omega \mathrm{x}\Omega)$$O(\lambda^{-q})$$arrow\infty$$s(\omega,\omega’;\lambda)$$S_{0}(\lambda)$$s_{0}(\omega,\omega’;\lambda)$$S_{0}(\lambda)$$s_{0}(\omega,\omega’;\lambda)$$s(\omega,\omega’;\lambda)=s(-\omega’, -\omega;\lambda)$

as kernel of the SM itself. Kernel (1.12) is also gauge invariant. This means that, for, the integrand in (1.12) is not changed if the functions
asmooth function
arereplaced byand the magnetic potential is replaced by. We emphasize
however that throughout the paper we do not use any particular gauge.
$\varphi(x)$$u_{\pm}$$A$$e^{-:}\varphi u_{\pm}$$A-\nabla\varphi$

Formula (1.12) gives the singular part of the scattering amplitude off any neighborhoodof the hyperplane. Sinceis arbitrary, this determines the singular part offor all. We note that the leading diagonal singularity of
was found in [23] for $\rho_{v}\in(1/2,1]$ and $A=0$ .
$\omega_{0}\in \mathrm{S}^{d-1}$$\Pi_{\omega_{0}}$$\omega,\omega’\in \mathrm{S}^{d-1}$$s(\omega,\omega’;\lambda)$$s(\omega,\omega’, \lambda)$

4. Our approach to the proof of formula (1.12) relies (even in the short-range case)on the expression of the SM via modified wave operators$W_{\pm}(H, H_{0};J_{\pm})=s- \lim_{tarrow\pm\infty}e^{:Ht}J_{\pm}e^{-*H_{0}t}$,(1.14)

where PDO
are constructed in terms of the functions. Following [9], we killneighborhoods of “bad” directions
by appropriate cut-0ff functions. Let$u_{\pm}(x,\xi)$$J_{\pm}$$\hat{x}=\mp\hat{\xi}$$\zeta_{\pm}(x,\xi)$(1.15)$T_{\pm}=HJ_{\pm}-J_{\pm}H_{0}$be the “effective” perturbation. The SM
corresponding to wave operators (1.14)
admits (see [10, 23, 24, 19]) the representation$S(\lambda)$$S(\lambda)=S_{1}(\lambda)+S_{2}(\lambda)$,(1.16)

where(1.17)$S_{1}(\lambda)=-2\pi i\Gamma_{0}(\lambda)J_{+}^{*}T_{-}\Gamma_{0}^{*}(\lambda)$

and$5_{2}(\mathrm{A})$$=2\pi i\Gamma_{0}(\lambda)T_{+}^{*}R(\lambda+i0)T_{-}\Gamma_{0}^{*}(\lambda)$.

(1.18)

Both these expressions are correctly defined which will be discussed in Sections 5and 4,
respectively.
With the help of the so called propagation estimates [17, 12, 11] we show in Section 4that the operator
has smooth kernel rapidly decaying as. Therefore we$S_{2}(\lambda)$$\lambdaarrow\infty$$\mathrm{c}\mathrm{a}1$

59
is given by explicit expression. The singular partthe regular part of the$H$(1.17) not depending on the resolvent of the operator . However it contains the cut. In Section 5we get rid of these auxiliary functions and, neglecting
ofl functions
to the, transform the kernel of-kernels decaying faster than any power ofinvariant expression (1. 12).
$S_{1}(\lambda)$$\mathrm{S}\mathrm{M}$$S_{2}(\lambda)$$\zeta_{\pm}$$\lambda^{-1}$$C^{\infty}$$S_{1}(\lambda)$

THE EIKONAL AND TRANSPORT EQUATIONS

2.

In this section we give astandard construction of approximate but explicit solutions of the
Schrodinger equation. This construction relies on asolution of the corresponding eikonal
and transport equations by iterations.
1. Let us plug expression (1.9) into the Schrodinger equation (1.8). Then$(-\Delta+2i\langle A(x), \nabla\rangle+V_{1}(x)-|\xi|^{2})(e^{i\Theta}\mathrm{b})$$=e^{i\Theta}(|\nabla|^{2}\mathrm{b}-i(\Delta)\mathrm{b}-2i\langle\nabla, \nabla \mathrm{b}\rangle-\Delta \mathrm{b}$-2$\langle A, \nabla\rangle \mathrm{b}+2i\langle A, \nabla \mathrm{b}\rangle+V_{1}\mathrm{b}-|\xi|^{2}\mathrm{b})$We require that the phase $(x, \xi)$ and the amplitude
the eikonal and transport equations, that is,$\nabla=\nabla_{x}$$\mathrm{b}(x, \xi)$

.

(2.1)

be approximate solutions of$|\nabla|^{2}-2\langle A,\nabla\rangle+V_{0}-|\xi|^{2}=\mathrm{q}_{0}(x,\xi)$

(2.2)

,

and
$-2i\langle\nabla, \nabla \mathrm{b}\rangle+2i\langle A, \nabla \mathrm{b}\rangle-\Delta \mathrm{b}+(-\mathrm{i}\mathrm{A}9+i\mathrm{d}\mathrm{i}\mathrm{v}A+\mathrm{q}_{0})\mathrm{b}=r(x, \xi)$

.

(2.3)

It follows from (2.1) that, for such fuctions $$ and , equality (1.10) is satisfied with thesame function $r(x, \xi)$ as in (2.3). When considering (2.2), (2.3), we always remove either(for the sign(for the sign $”+”$ ) or
aconical neighborhood of the direction
defined by“-,,). We choose $(x, \xi)=_{\pm}(x, \xi)$ in such away that
. Then we construct(2.2) is ashort-range function of , and it tends to 0as
so that $r(x, ()$ $=r_{\pm}(x, \xi)$ decays as $|x|arrow\infty$ as an arbitrary given.
. It turns out that $r(x, \xi)$ has asimilar decay also in the variable
power ofand consider the
If $V$ is short-range and $A=0$ , then we can set
necessary
for any(2.2)
istransport equation (2.3) only. However, the eikonal equation
non-trivial magnetic potential or (and) long-range electric potential . The transport
equation is always unavoidable because, as we shall see below, the functionin (1.9) is not
only and hence, for example, the choice
at infinity as
sufficient.
$\mathrm{b}$$\hat{x}=\hat{\xi}$$\hat{x}=-\hat{\xi}$$\mathrm{q}_{0}(x, \xi)=\mathrm{q}_{0}^{(\pm)}(x, \xi)$$|\xi|arrow\infty$$x$$\mathrm{b}(x, \xi)=\mathrm{b}_{\pm}(x, \xi)$$|\xi|^{-1}$$|x|^{-1}$$_{\pm}(x, \xi)=\langle x, \xi\rangle$$V$$\Delta\pm \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}\mathrm{s}$$\mathrm{b}_{\pm}=1$$|x|^{-1-\rho}$

We seek $_{\pm}(x, \xi)$ in the form$_{\pm}(x, \xi)=\langle x, \xi\rangle+\Phi_{\pm}(x, \xi)$

where

,

(2.4)

tends to zero as $|x|arrow\infty$ off any conical neighborhood of the direction. We construct asolution of equation (2.2) by iterations. Actually, we set$(\nabla\Phi_{\pm})(x, \xi)$$\hat{x}=\mp\hat{\xi}$$\Phi_{\pm}(x, \xi)=\Phi_{\pm}^{(N_{0})}(x,\xi)=\sum_{n=0}^{N_{0}}(2|\xi|)^{-n}\phi_{n}^{(\pm)}(x,\hat{\xi})$

(2.5)

60
and plug expressions (2.4) and (2.5) into equation (2.2). Comparing coefficients at the$N_{0}-1$ , we obtain the equations, n $=-1,$ 0,

same powers of$(2|\xi|)^{-n}$\ldots ,$\langle\hat{\xi}, \nabla\phi_{0}\rangle=\langle\hat{\xi},$

A\rangle ,$\langle\hat{\xi}, \nabla\phi_{1}\rangle+|\nabla\phi_{0}|^{2}-2\langle A, \nabla\phi_{0}\rangle+V_{0}=0$$\langle\hat{\xi}, \nabla\phi_{n+1}\rangle+\sum_{m=0}^{n}\langle\nabla\phi_{m}, \nabla\phi_{n-m}\rangle-2\langle A, \nabla\phi_{n}\rangle=0$,$n\geq 1$

,

(2.6)

,

(2.7)

so that the “error term” equals.

$\mathrm{q}_{0}(x,\xi)=\sum_{n+m\geq N_{0}}(2|\xi|)^{-n-m}\langle\nabla\phi_{n}, \nabla\phi_{m}\rangle-2(2|\xi|)^{-N_{0}}\langle A, \nabla\phi_{N_{0}}\rangle$

All equations (2.6), (2.7) have the form
(2.8)$\langle\hat{\xi}, \nabla\phi(x,\hat{\xi})\rangle+f(x,\hat{\xi})=0$

and can be explicitly solved. Let the domain
be distinguished by thecondition:
if either $|x|\leq R$ orfor some. Of
course, all constants below depend on and $R$ . The following assertion is almost obvious(see [23], for details).
$\Gamma_{\pm}(\epsilon, R)\subset \mathrm{R}^{d}\chi$$(x, \langle)$$\mathrm{R}^{d}$$\pm\langle\hat{x},\hat{\xi}\rangle\geq-1+\epsilon$$\in\Gamma_{\pm}(\epsilon, R)$$\epsilon>0$$\epsilon$

Lemma 2.1 Suppose that
(2.9)$|\partial_{x}^{\alpha}d_{\xi}f(x,\hat{\xi})|\leq C_{\alpha,\beta}|\xi|^{-|\beta|}(1+|x|)^{-\rho-|\alpha|}$for$x\in\Gamma_{\pm}(\epsilon, R)$

and some $\rho>1$ . Then the function(2.10)$\phi^{(\pm)}(x,\hat{\xi})=\pm\int_{0}^{\infty}f(x\pm t\hat{\xi},\hat{\xi})dt$

satisfies equation (2.8) and the estimates,$|\mathfrak{X}_{x}ff_{\zeta}i_{\phi^{(\pm)}(x,\hat{\xi})|}\leq C_{\alpha,\beta}|\xi|^{-|\beta|}(1+|x|)^{1-\rho-|\alpha|}$$x\in\Gamma_{\pm}(\epsilon, R)$.

(2.11)

If estimates (2.9) are fulfilled for some $\rho\in(0,1)$ $on/y$, then the function(2.12)$\phi^{(\pm)}(x,\hat{\xi})=\pm\int_{0}^{\infty}(f(x\pm t\hat{\xi},\hat{\xi})-f(\pm t\hat{\xi},\hat{\xi}))dt$

satisfies both equation (2.8) and estimates (2.11).
Proceeding by induction, we can solve by formulas (2.10) or (2.12) all equations (2.6)
and (2.7). The case where $V$ and are both short-range is discussed specially in subs. 3.
$A$

Here we focus on the long-range case. Let us formulate the corresponding result.

Proposition 2.2 Let assumption (1.7) hold for some $\rho\in(0,1)$ . Then estimates$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\phi_{n}^{(\pm)}(x,\hat{\xi})|\leq C_{\alpha,\beta}|\xi|^{-|\beta|}(1+|x|)^{1-n\rho-|\alpha|}$,$n=1,2$ ,$\ldots$,and$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\mathrm{q}_{0}^{(\pm)}(x, \xi)|\leq C_{\alpha,\beta}|\xi|^{-N_{0}-|\beta|}(1+|x|)^{-N_{0}\rho-|\alpha|}$are fulfilled on the setR) for all multi-indicesthesameestimateas.
satisfies$\Gamma_{\pm}(\epsilon,$$\phi_{1}^{(\pm)}(x,\hat{\xi})$$\alpha$and$\beta$.

. The function$\phi_{0}^{(\pm)}(x,\hat{\xi})$

61
Corollary 2.3 The function (2.5) satisfies the estimates$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\Phi_{\pm}(x, \xi)|\leq C_{\alpha,\beta}|\xi|^{-|\beta|}(1+|x|)^{1-\rho-|\alpha|}$,$x\in\Gamma_{\pm}(\epsilon, R)$.

(2.13).
in (2.4) is subject to the only restriction
Below the number
Of course, in particular cases the procedure above can be simplified. For example, if$A=0$ andis long-range but $\rho_{v}>1/2$ , then$N_{0}\rho\geq 2$$N_{0}$$V$$\Phi^{(\pm)}(x, \xi)=(2|\xi|)^{-1}\phi_{1}^{(\pm)}(x,\hat{\xi})=\pm 2^{-1}\int_{0}^{\infty}(V(x\pm t\xi)-V(\pm t\xi))dt$.

2. An approximate solution of the transport equation (2.3) can be constructed by aprocedure similar to the one given above. Using (2.4), we rewrite this equation as$-2\mathrm{i}$(A$\nabla \mathrm{b}\rangle$(2.14)$+2\mathrm{i}(\mathrm{A}-\nabla\Phi, \nabla \mathrm{b}\rangle-\Delta \mathrm{b}+(-i\Delta\Phi+\mathrm{i}\mathrm{d}\mathrm{i}\mathrm{v}\cdot 4+\mathrm{q}_{0})\mathrm{b}=r.$in the form (1.4) with bounded in 4coefficients
We look for the function(2.14),
we obtain the following recurrent equations
Plugging this expression into$b_{n}^{(\pm)}(x, ()$$\mathrm{b}_{\pm}(x, \xi)$,$\langle\hat{\xi}, \nabla b_{n+1}\rangle=2i\langle A-\nabla\Phi, \nabla b_{n}\rangle-\Delta b_{n}+(-i\Delta\Phi+i\mathrm{d}\mathrm{i}\mathrm{v}A+\mathrm{q}_{0})b_{n}$$n=0,1$ ,$\ldots$.

, $N-1$ . (2.15)

Then$r(x, \xi)=r^{(N)}(x, \xi)=(2i|\xi|)^{-N}\langle\hat{\xi}, \nabla b_{N+1}\rangle$

All these equations have the form (cf. (2.8))$\langle\hat{\xi}, \nabla b_{n+1}(x, \xi)\rangle+f_{n}(x, \xi)=0$,, . Therefore they can be solved bydepends on ,where ashort-range function
2.1 we obtain
Lemma
again
using
one of the formulas (2.10). Thus,$b_{1}$$f_{n}$$\ldots$$b_{n}$

Proposition 2.4 Let assumption (1.7) hold, let $\rho_{1}=\min\{1, \rho\}$ and let, $n\geq 1$ , satisfy the estimates
Then functions$(x, \xi)\in\Gamma_{\pm}(\epsilon, R)$.

$b_{n}^{(\pm)}$$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}b_{n}^{(\pm)}(x, \xi)|\leq C_{\alpha,\beta}|\xi|^{-|\beta|}(1+|x|)^{-\rho_{1}n-|\alpha|}$.

The right-hand side of equation (2.3) satisfies$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}r_{\pm}^{(N)}(x, \xi)|\leq C_{\alpha,\beta}|\xi|^{-N-|\beta|}(1+|x|)^{-1-\rho_{1}(N+1)-|\alpha|}$.

(2.16)

Corollary 2.5 The function (1.4) satisfies the estimates$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\mathrm{b}_{\pm}(x, \xi)|\leq C_{\alpha,\beta}|\xi|^{-|\beta|}(1+|x|)^{-|\alpha|}$.

(2.17)

Combining Propositions 2.2 and 2.4, we get the final result.

constructed in Propositions 2.2
andTheorem 2.6 For the functions$u_{\pm}(x, \xi)=u_{\pm}^{(N)}(x, \xi)$and 2.4, respectively, and for the functions
defined by (1.9), equalsatisfying estimates (2.16) in the regio
holds with the remainder$_{\pm}^{(N_{0})}(x, \xi)$$r_{\pm}^{(N)}(x, \xi)$$-J-\backslash ity(1.10)$$\mathrm{b}_{\pm}^{(N)}(x, \xi)$

$n$

62
We emphasize that in contrast to the parameter

3. Of course, the functions
depend on
the short-range caseand equation (2.15) reduces to$N_{0}$

which is fixed, we need $Narrow\infty$ .

. However, incontain different powers of$A=0$. Then $\Phi=0$
and only. Suppose first that$b_{n}^{(\pm)}(x, \xi)$$b_{n}^{(\pm)}$$x$$|\xi|^{-1}$$\hat{\xi}$$\langle\hat{\xi}, \nabla b_{n+1}\rangle=-\Delta b_{n}+Vb_{n}$.

Thus, we obtain the folowing assertion.

Proposition 2.7 Let $A=0$ and letsatisfy assumption (1.7) with $\rho_{v}>1$ . Let
the sum (1.4) and the functions
where
are definedby recurrent formulas
and$V$$u\pm(x, \xi)=e^{:(x,\xi)}\mathrm{b}_{\pm}(x, \xi)$$b_{n}^{(\pm)}(x,\hat{\xi})$$\mathrm{b}_{\pm}is$$b_{0}^{(\pm)}=1$$b_{n+1}^{(\pm)}(x, \hat{\xi})=\mp\int_{0}^{\infty}(-\Delta b_{n}^{(\pm)}(x\pm t\hat{\xi},\hat{\xi})+V(x\pm t\hat{\xi})b_{n}^{(\pm)}(x\pm t\hat{\xi},\hat{\xi}))dt$

Then for$(x,\xi)\in\Gamma_{\pm}(\epsilon, R)$.

and $\rho_{2}=\min\{2,\rho_{v}\}$(2.18)$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}b_{n}^{(\pm)}(x,\hat{\xi})|\leq C_{\alpha,\beta}|\xi|^{-|\beta|}(1+|x|)^{-(\rho_{2}-1)n-|\alpha|}$and the remainder (1.10) satisfies the estimates$|\partial_{x}^{\alpha}\partial_{\xi}^{\theta}r_{\pm}^{(N)}(x, \xi)|\leq C_{\alpha},\rho|\xi|^{-N-|\beta|}(1+|x|)^{-(\rho_{2}-1)(N+1)-|\alpha|}$Let us write down explicit expressions for the first two functions$b_{1}^{(\pm)}(x, \hat{\xi})=\mp\int_{0}^{\infty}V(x\pm t\hat{\xi})dt$$b_{n}$.

(2.19):,$b_{2}^{(\pm)}(x, \hat{\xi})=-\int_{0}^{\infty}t(\Delta V)(x\pm t\hat{\xi})dt+\frac{1}{2}(\int_{0}^{\infty}V(x\pm t\hat{\xi})dt)^{2}$.

If amagnetic potential is non-trivial, then,$\Phi_{\pm}(x,\hat{\xi})=\phi_{0}^{(\pm)}(x,\hat{\xi})=\mp\int_{0}^{\infty}\langle\hat{\xi}$and$\mathrm{q}_{0}^{(\pm)}=|\nabla\Phi_{\pm}|^{2}-2\langle A, \nabla\Phi_{\pm}\rangle+V_{0}$Hence it follows ffom (2.15) that the coefficientsand(2.20)$A(x\pm t\hat{\xi})\}dt$$b_{n}^{(\pm)}(x,\hat{\xi})$.
are determined by formulas$b_{0}^{(\pm)}=1$$b_{n+1}^{(\pm)}(x, \hat{\xi})=\mp\int_{0}^{\infty}f_{n}^{(\pm)}(x\pm t\hat{\xi},\hat{\xi})dt$,(2.21)

where$f_{n}^{(\pm)}=2i\langle A-\nabla\Phi_{\pm}, \nabla b_{n}^{(\pm)}\rangle-\Delta b_{n}^{(\pm)}$$+(|\nabla\Phi_{\pm}|^{2}-2\langle A, \nabla\Phi_{\pm}\rangle+V_{1}-i\Delta\Phi_{\pm})b_{n}^{(\pm)}$.

(2.22)Let us formulate the result obtained.
$= \min\{2,\rho\}$ .
Proposition 2.8 Let and satisfy assumption (1.7) with $\rho>1$ , and(2.4)(2.20).
andby reLetthebeconstructedbyDefinefunctionsformulas$T/ien$current formulas (2.21), (2.22) and let
be the sum (1.4).
estimates (2.18) onand (2.19) on the remainder (1.10) hold.
$A$

$V$

$let\hslash$$\Theta(x,\xi)$$b_{n}^{(\pm)}$$\mathrm{b}_{\pm}$$b_{n}^{(\pm)}$63
3.

WAVE OPERATORS AND THE SCATTERING MATRIX

1. Let us recall briefly some basic facts about PDO (see, e.g., [6] or [20]). Let(Af)(x)$=(2 \pi)^{-d/2}\int_{\mathrm{R}^{d}}e^{i\langle x,\xi\rangle}a(x, \xi)\hat{f}(\xi)d\xi$,

and thewhereis the Fourier transform of from, say, the Schwartz space. Sometimes it is more convenient to consider more general PDO
symboldetermined by their amplitudes. We define such operators in terms of the corresponding
sesquilinear forms$\hat{f}=\mathcal{F}f$$S(\mathrm{R}^{d})$$f$$a\in C^{\infty}(\mathrm{R}^{d}\cross \mathrm{R}^{d})$

(3.1)

,

$( \mathrm{A}f, g)=(2\pi)^{-d}\int_{\mathrm{R}^{d}}\int_{\mathrm{R}^{d}}\int_{\mathrm{R}^{d}}e^{i\langle x,\xi’-\xi\rangle}\mathrm{a}(x, \xi, \xi’)\hat{f}(\xi’)\overline{\hat{g}(\xi)}d\xi d\xi’dx$

where the amplitude$\mathrm{a}(x, \xi, \xi’)$is also a$C^{\infty}$-function of all its variables.

It is standard to assume that and abelong to Hormander classes. Set $(x)=(1+$,. By definition, the symbol (or the corresponding operator$A)$ belongs to the class, if for all multi-indices and, $\rho>0$ ,$a$$|x|^{2})^{1/2}$$\langle\xi\rangle=(1+|\xi|^{2})^{1/2}$$a$$\delta<1$$S^{n,m}(\rho, \delta)$$\beta$$\alpha$.

$|(\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a)(x, \xi)|\leq C_{\alpha,\beta}\langle x\rangle^{n-|\alpha|\rho+|\beta|\delta}\langle\xi\rangle^{m-|\beta|}$

into itself. For the
The operators from these classes send the Schwartz spacedependenceonamplitudes awe do not have to keep track of the4and . Thus,and , $\xi’\in K$
if for all multi-indices , , , any compact set$A$$S(\mathrm{R}^{d})$$\xi’$$S^{n}(\rho, \delta)$$\alpha$$\beta$$K\subset \mathrm{R}^{d}$$\beta’$$|(\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\partial_{\xi}^{\beta’},\mathrm{a})(x, \xi, \xi’)|\leq C_{\alpha,\beta,\beta’}(K)\langle x\rangle^{n-|\alpha|\rho+(|\beta|+|\beta’|)\delta}$$\mathrm{a}\in$

$\xi$

.

Under this assumption the form (3.1) is well-defined as an oscillating integral for. We omit in notation and if $\rho=1$ and $\delta=0$ .
Actually, we need amore special class of PDO with oscillating symbols$\hat{f},\hat{g}\in$$\delta$$C_{0}^{\infty}(\mathrm{R}^{d})$$\rho$$a(x, \xi)=e^{i\Phi(x,\xi)}\alpha(x, \xi)$

(3.2)

,

the class of symbols. We denote by, $\rho\in(0, 1)$ , andwhere (I)in the caseor operators obeying the conditions above. The definition of the class$C^{n,m}(\Phi)\subset
S^{n,m}(\rho,
1-\rho)$,thestandard
Since
quite
similar.
of oscillating amplitudes is
if $\rho>1/2$ . In the general case the oscillating
PDO calculus works in the classes
factors $\exp(i\Phi(x, \xi))$ or $\exp(i\Phi(x, \xi, \xi’))$ should be explicitly taken into account.
The proof of the following assertion can be found either in [13] or [25]. We often
andfor the operators of multiplication by these functions in theuse the notation
respectively.
representations,andmomentumcoordinate$\in S^{1-\rho,0}$$\alpha\in S^{n,m}$$C^{n,m}(\Phi)$$C^{n}(\Phi)$$C^{n,m}(\Phi)$$\langle x\rangle$$\langle\xi\rangle$

Proposition 3.1 Let a $\in C^{n,m}(\Phi)$ , n.
bounded in the space$\leq 0$and m$\leq 0$. Then the operator$L_{2}(\mathrm{R}^{d})$This result extends naturally to PDO defined by (3.1)$A\langle x\rangle^{-n}$is64
We need also aclasssymbols such that $a(x,$ () $=0$ if$>0$ . Moreover, we assume that $a(x,$() $=0$ if
orclass. Then we set

for some
for symbols from this$–\pm-\mathrm{o}\mathrm{f}$$|x|\leq\epsilon$$\epsilon$$S_{\pm}^{n,m}( \rho, \delta)=S^{n,m}(\rho, \delta)\bigcap_{-\pm}^{-}-$,$\mp\langle\hat{x}$$|\xi|\leq\epsilon$$C_{\pm}^{n,m}( \Phi)=C^{n,m}(\Phi)\bigcap_{-\pm}^{-}-$, ()$\leq\epsilon$.

2. Let $H_{0}=-\Delta$ and the operator $H$ defined by (1.6) act in the space $H$.
Denote by
and their spectral projections. Note that, as shown in $[8, 22]$ where theproof of [18] was extended to magnetic potentials, the operator $H$ does not have positive
eigenvalues. In the long-range case the wave operators (1.14) exist only for aspecial choiceof identifications . We construct
as PDO.
Let, $\gamma>1$ , be such that $\sigma(\tau)=1$ if $\tau\in[-1,1-2\epsilon]$ for some$\epsilon\in(0, 1/2)$ and $\sigma(\tau)=0$ if $\tau\in[1-\epsilon, 1]$ .
Let y7be such that $\eta(x)=0$ in
aneighborhood of zero and $\eta(x)=1$ for large. We denote by afunction
which equals to zero in aneighborhood of 0and
for, say,(for some
. Set
.
Let
be the function (it depends on
and $N$ ) defined in the previous section(see Theorem 2.6). Following [9], we construct
by the formula$=L_{2}(\mathrm{R}^{d})$$E_{0}$$E$$J_{\pm}$$J_{\pm}$$\sigma\in C^{\infty}(-\gamma,\gamma)$$\in C^{\infty}(\mathrm{R}^{d})$$|x|$$\theta$$\theta(\lambda)=1$$\lambda_{0}>0)$$C^{\infty}(\mathrm{R}_{+})$$\lambda\geq\lambda_{0}$$\zeta_{\pm}(x,\xi)=\sigma(\mp\eta(x)\langle\hat{\xi},\hat{x}\rangle)\theta(|\xi|^{2})$$u_{\pm}(x,\xi)$$N_{0}$$J_{\pm}$$(J_{\pm}f)(x)=(2 \pi)^{-d/2}\int_{\mathrm{R}^{d}}u_{\pm}(x,\xi)\zeta_{\pm}(x,\xi)\hat{f}(\xi)d\xi$

.

(3.3)

Thus,
is aPDO with symbol (3.2) whereand. Weemphasize however that in contrast to [9] the symbolof the operator
is quitean explicit function. This is essential for construction of the asymptotic expansion of the. Due to the cut-0ff functions
and estimates (2.17) on, we have that
. The function
is of course singular on the set
but satisfies theestimates of the classon the support of . Abusing somewhat terminology, we
write. By Proposition 3.1, the operator is bounded.
It is shown in [9, 23, 19] that the wave operators (1.14) exist which implies the intertwining property $W_{\pm}(H, H_{0\backslash }..J_{\pm})H_{0}=HW_{\pm}(H, H_{0;}J_{\pm})$ . Moreover, they are isometric. on
the subspace
and are complete, that is$\Phi=\Phi_{\pm}$$J_{\pm}$$\alpha_{\pm}(x,\xi)=\mathrm{b}_{\pm}(x,\xi)\zeta_{\pm}(x,\xi)$$a_{\pm}(x,\xi)$$\mathrm{S}\mathrm{M}$$J_{\pm}$$\zeta\pm(x, \xi)$$\alpha\pm\in S^{0,0}$$\mathrm{b}_{\pm}(x, \xi)$$\Phi_{\pm}(x,\xi)$$\hat{x}=\mp\hat{\xi}$$S^{1-\rho,0}$$\zeta_{\pm}$$J_{\pm}\in C^{0,0}(\Phi_{\pm})$$J_{\pm}$$E_{0}(\lambda_{0}, \infty)H$

Ran$(W_{\pm}(H, H_{0;}J_{\pm})E_{0}(\lambda_{0}, \infty))=\mathrm{a}(\mathrm{x}, \infty)H$.

In the short-range case$s- \lim_{tarrow\pm\infty}(J_{\pm}-\theta(H_{0}))e^{-\dot{|}H_{0}t}=0$,

so that the wave operators
coincide with the usual wave operators $W_{\pm}(H, H_{0})$(times). The scattering operator is defined by the standard relation$W_{\pm}(H, H_{0};J_{\pm})$$\theta(H_{0})$$\mathrm{S}=\mathrm{S}$( $H$ , Ho;

It commutes with the operator$J_{+},$ $J_{-}$$H_{0}$) $=W_{+}^{*}(H, H_{0;}J_{+})W_{-}$ ( $H$, Ho;and is unitary on the space$J_{-}$).

$E_{0}(\lambda_{0}, \infty)H$.

3. Let us calculate the perturbation (1.15). According to (1.10), we have that$g\pm(x,\xi):=(-\Delta+2i\langle A(x), \nabla\rangle+V_{1}(x)-|\xi|^{2})(u_{\pm}(x, \xi)\zeta_{\pm}(x, \xi))$$=q\pm(x,\xi)\zeta_{\pm}(x,\xi)-2\langle\nabla u_{\pm}(x, \xi), \nabla\zeta_{\pm}(x,\xi)\rangle$$-u\pm(x,\xi)(\Delta\zeta\pm)(x,\xi)+2iu\pm(x,\xi)\langle$ $A(x)$, Vx (x)$\xi)\rangle$

.

(3.3)

65
Now it follows from (3.3) that$(T_{\pm}f)(x)$where$t_{\pm}^{(r)}=\exp(i\Phi_{\pm})\tau_{\pm}^{(r)}$$\tau_{\pm}^{(r)}=r_{\pm}\zeta_{\pm}$,$=$$(2 \pi)^{-d/2}\int_{\mathrm{R}^{d}}g_{\pm}(x, \xi)\hat{f}(\xi)d\xi$$=$$(2 \pi)^{-d/2}\int_{\mathrm{R}^{d}}e^{i\langle x,\xi\rangle}(t_{\pm}^{(r)}(x, \xi)+t_{\pm}^{(s)}(x, \xi))\hat{f}(\xi)d\xi$$=$: $(T_{\pm}^{(r)}f)(x)+(T_{\pm}^{(s)}f)(x)$ ,,$t_{\pm}^{(s)}=\exp(i\Phi_{\pm})\tau_{\pm}^{(s)}$

(3.5)

and

$\tau_{\pm}^{(s)}=-2i\mathrm{b}_{\pm}\langle\xi+\nabla\Phi_{\pm}-A, \nabla\zeta_{\pm}\rangle-2\langle\nabla \mathrm{b}_{\pm}, \nabla\zeta\pm\rangle-\mathrm{b}\pm\Delta\zeta\pm\cdot$

Due to the cut-0ff functions
Propositions 2.2 and 2.4.

$\zeta_{\pm}$,$\nabla\zeta_{\pm}$and$\Delta\zeta_{\pm}$, the next result follows directly from

Proposition 3.2 Let assumption (1.7) hold and let $\rho_{1}=\min\{1, \rho\}$ . Thenand$t_{\pm}^{(r)}\in C^{-1-\rho_{1}(N+1),-N}(\Phi_{\pm})$$t_{\pm}^{(s)}\in C_{\pm}^{-1,1}(\Phi_{\pm})$.

, let the operator
4. Let:be defined by formula (1.1)
and let $(Uf)(\lambda)=\Gamma_{0}(\lambda)f$ . Then
: $H$
extends by continuity to aunitary operator andas multiplication by the independentacts in the spaceas multiplication, the operator
variable A. Since
acts in the space.
by the operator-function:
known as thein the case where
We need astationary formula (see [10, 23, 24, 19]) for theidentifications
andfor $tarrow+\infty$ and $tarrow-\infty$ are different. Since auxiliary wave
operators,$\sigma \mathfrak{n}$$=L_{2}(\mathrm{S}^{d-1})$$S(\mathrm{R}^{d})arrow 9\mathrm{t}$$\Gamma_{0}(\lambda)$$U$$arrow\tilde{H}=L_{2}(\mathrm{R}_{+}; \mathfrak{R})$$UH_{0}U^{*}$$\tilde{H}$$\mathrm{S}H_{0}=H_{0}\mathrm{S}$$U\mathrm{S}U^{*}$$S(\lambda)$$\mathfrak{R}$$\tilde{H}$$\mathrm{S}\mathrm{M}$$arrow \mathfrak{R}$$\mathrm{S}\mathrm{M}S(\lambda)$$J_{+}$$J_{-}$$s- \lim_{tarrow\pm\infty}e^{iH_{0}t}J_{+}^{*}J_{-}e^{-iH_{0}t}=0$

we have the following result.
Proposition 3.3 Let assumption (1.7) hold. Then the $SM$ admits the representation(1.16) whereandare given by formulas (1.17) and (1.18), respectively.
$S_{1}(\lambda)$$A$$S_{2}(\lambda)$

Let us discuss here the precise meaning of the expression
is an operator acting on functions defined on . Put$A^{\mathrm{b}}(\lambda):=\Gamma_{0}(\lambda)A\Gamma_{0}^{*}(\lambda)$

where$\mathrm{R}^{d}$$\delta_{\epsilon}(|\xi|^{2}-\lambda)=\epsilon\pi^{-1}((|\xi|^{2}-\lambda)^{2}+\epsilon^{2})^{-1}$,and letbe an arbitrary function such that $\gamma_{j}(k)=1$ . Taking into account, ) forby(1.1), we define (see, e.g., [24]) the sesquilinear form (Athe relation$\gamma_{j}\in C_{0}^{\infty}(\mathrm{R}_{\dagger})$$(\lambda)w_{1}$$(A^{\mathrm{b}}(\lambda)w_{1}, w_{2})$where$k=\lambda^{1/2}$$w_{2}$$w_{j}\in C^{\infty}(\mathrm{S}^{d-1})$$=2k^{-d+2} \lim_{\epsilonarrow 0}(A\mathcal{F}^{*}\delta_{\epsilon}(|\xi|^{2}-\lambda)\hat{\psi}_{1}, \mathcal{F}^{*}\delta_{\epsilon}(|\xi|^{2}-\lambda)\hat{\psi}_{2})$

,

(3.6)

,
$\hat{\psi}_{j}(\xi)=w_{j}(\hat{\xi})\gamma_{j}(|\xi|)$,$j=1,2$ ,is well defined if
provided the limit in the right-hand side exists. The form
is an
. This is, of course, true if
the limit (3.6) exists for all$(A^{\mathrm{b}}(\lambda)w_{1}, w_{2})$$w_{j}\in C^{\infty}(\mathrm{S}^{d-1})$$G=\mathcal{F}A\mathcal{F}^{*}$

66
integral operator with kernel
which is continuous near the surface $|\xi|=|\xi’|=k$ .
In this caseis also an integral operator onwith kernel$G(\xi, \xi’)$$\mathrm{S}^{d-1}$$A^{\mathrm{b}}(\lambda)$$g(\omega,\omega’;\lambda)=2^{-1}k^{d-2}G(k\omega, k\omega’)$

.

(3.7)

Furthermore, by the Sobolev trace theorem, limit (3.6) exists and hence the operator
is well-defined (and is bounded in the space) if$L_{2}(@^{d-1})$$A^{\mathrm{b}}(\lambda)$

A

(3.8)

$=\langle x\rangle^{-\mathrm{r}}B\langle x\rangle^{-r}$for abounded operator B in
and r $>1/2$ . This means that the operators
are also well-defined for PDO Aof order n $<-1$ .
We note that the stationary representation of the SM is determined exactly by thelimits as the one in the right-hand side of (3.6).
$L_{2}(\mathrm{R}^{d})$$\mathrm{A}^{\mathrm{b}}(\lambda)$

To estimate in the next section the regular part
obvious remark.

$S_{2}(\lambda)$of the SM, we need the following

Lemma 3.4 Suppose that (3.8) is satisfiedfor $r>d/2$ . Set
Then the operator
has continuous kernel$u_{0}(x,\omega, \lambda)=\exp(i\lambda^{1/2}\langle\omega, x\rangle)$.

$A^{\mathfrak{y}}(\lambda)$$g(\omega,\omega’;\lambda)=2^{-1}k^{d-2}(2\pi)^{-d}(B\langle x\rangle^{-\mathrm{r}}u_{0}(\omega’, \lambda), \langle x\rangle^{-f}u_{0}(\omega, \lambda))$Moreover, this function belongs to the classis bounded by $Ck^{d-2+p}$ .

$C^{p}(\mathrm{S}^{d-1}\cross \mathrm{S}^{d-1})$$C^{p_{-}}nom$.

for $p<r-d/2$ and its

To treat the singular part
, we apply definition (3.6) to the PDO
determined by its amplitude. In this case, by Proposition 3.2, ais of,and hence the operators
are defined only under special assumptions on . According
to (3.4), (3.5), up to an integral operator with smooth kernel, Ahas the amplitude which,due to the functions, equals zero if
andis close to 1or -1(in aneighborhood of the conormal bundle of each sphere $|\xi’|=k$ ). In this case, as
shown in [25], the operators
are correctly defined by formula (3.6) in aspace of(the case of PDO determined by their symbols was considered earlier infunctions on[15] . Moreover, they are also PDO, and an explicit expression for their amplitudes wasgiven in [25]. However, our construction of the singular part of the scattering matrix in
Section 5is, at least formally, independent of the results of [25]. It is important that thisconstruction allows us to get rid of the cut-0ff functions
and to obtain an arbitraryclose approximation to the.
$S_{1}(\lambda)$$\mathrm{A}=J_{+}^{*}T_{-}$$\mathrm{a}(x, \xi, \xi’)$$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}-1$$\mathrm{A}^{\mathrm{b}}(\lambda)$$\mathrm{a}$$\nabla\zeta_{-}(x,\xi’)$$\Delta\zeta_{-}(x, \xi’)$$\langle\hat{x},\hat{\xi}’\rangle$$\mathrm{A}^{\mathrm{b}}(\lambda)$$\mathrm{S}^{d-1}$$)$$\zeta_{\pm}$$\mathrm{S}\mathrm{M}$4.

THE REGULAR PARTIn this section we show that the regular part (1.18) of the SM is negligible.

1. Recall that the functions
were constructed in Theorem 2.6 and that thecorresponding operators
andwere defined by equations (3.3) and(3.5), respectively. Our main analytical result here is the following$u_{\pm}=u_{\pm}^{(N)}$$J_{\pm}=J_{\pm}^{(N)}$$T_{\pm}=T_{\pm}^{(N)}$67
Proposition 4.1 For any

$p$

and

$q$

there exists $N$ such that for$T_{\pm}=T_{\pm}^{(N)}$the operators$B_{p,q,N}(\lambda)=\langle x\rangle^{p}\langle\xi\rangle^{q}T_{+}^{*}R(\lambda+i0)T_{-}\langle\xi\rangle^{q}\langle x\rangle^{p}$

are bounded uniformly in$\lambda\geq\lambda_{0}>0$.

This result will be checked in the following subsections. Let us first of all show that it.
implies regularity of the operator$S_{2}(\lambda)$

Theorem 4.2 For any and there exists $N$ such that for(1.18) has kernel
which belongs to the class.
as A
of this kernel is$p$Remark that$arrow\infty$$\Gamma_{0}(\lambda)\langle\xi\rangle^{-q0}=(1+\lambda)^{-q\mathrm{o}/2}\Gamma_{0}(\lambda)$

S2 (X)$C^{p}$$C^{p}(\mathrm{S}^{d-1}\cross \mathrm{S}^{d-1})$$s_{2}(\omega, \omega’;\lambda)$$O(\lambda^{-q})$, the operator
and the-norrm

$T_{\pm}=T_{\pm}^{(N)}$$q$and hence$=2\mathrm{n}\mathrm{i}(1+\lambda)^{-q0}\Gamma_{0}(\lambda)\langle x\rangle^{-n)}B_{\infty,q0,N}(\lambda)\langle x\rangle^{-p0}\Gamma_{0}^{*}(\lambda)$.

Let $p_{0}>d/2+p$ and $q_{0}\geq q-1+(d+p)/2$ . We suppose here that $N=N(p_{0}, q_{0})$ isare bounded uniformly
the same as in Proposition 4.1, so that the operators
belongs to theoperator
the
3.4,
of
Then,
the
kernel
in A
in
Lemma
as shown.
$Ck^{d-2+p-2q0}$
which is estimated by, and its -norm is bounded byclass$Ck^{-2q}$. This concludes the proof of Theorem 4.2.
In the following subsections we shall give an idea of the proof of Proposition 4.1.
$B_{p0,q0,N}(\lambda)$$S_{2}(\lambda)$$\geq\lambda 0$$C^{p}$$C^{p}(\mathrm{S}^{d-1}\cross \mathrm{s}^{d-1})$

2. We need some results on the boundedness of combinations of PDO with symbols
with functions of the generator of(see subs. 1of Section 3) where
dilations$T$$\Phi\in S^{1-\rho,0}$$t\in C_{\pm}^{n,m}(\Phi)$$\mathrm{A}=\frac{1}{2}\sum_{j=1}^{d}(x_{j}D_{j}+D_{j}x_{j})$.

the spectral projection of the operator A.
First we formulate astrengthening of aresult of [11].

We denote by$\mathrm{p}_{\pm}=E_{\mathrm{A}}(\mathrm{R}\pm)$

Proposition 4.3 Let tk such that the operator$\in C_{\pm}^{n,m}(\Phi)$$\langle \mathrm{A}\rangle^{-k}T$for one of the signs and some n, m. Then there eists

is bounded.

Of course, this result is of interest only if at least one of the indices
The following assertion is also motivated by the results of [11].

$n$

or

$m$

is positive.

Proposition 4.4 Let t
for some n, m and $\rho>0$ , $\delta<1$ . Then the operator
is bounded for arbitrary p, q and k.
$\in S_{\pm}^{n,m}(\rho, \delta)$$\langle \mathrm{A}\rangle^{k}\mathrm{p}_{\pm}T\langle\xi\rangle^{q}\langle x\rangle^{p}$

The following resolvent estimates were deduced in [17, 12, 11] from the famous Mourre
estimate [16]. To obtain estimates at high energies, we use additionally the dilationtransformation.

Proposition 4.5 Let assumption (1.7) hold. Then for ${\rm Re} z>0$ ,functions(A)$-pR(z)\langle \mathrm{A}\rangle^{-p}$$\langle \mathrm{A}\rangle^{-1+\mathrm{p}2}\mathrm{P}_{-}R(z)\langle \mathrm{A}\rangle^{-\mathrm{P}1}$,,${\rm Im} z\geq 0$$p>1/2$ ,$\langle \mathrm{A}\rangle^{-p_{1}}R(z)\mathrm{P}_{+}\langle \mathrm{A}\rangle^{-1+p2}$

the operator(4.1)
(4.1)

68
for each $p_{1}>1/2$ ,and$p_{2}<p_{1}$(4.3)$\langle \mathrm{A}\rangle^{p}\mathrm{P}_{-}R(z)\mathrm{P}_{+}\langle \mathrm{A}\rangle^{p}$for arbitrary p are continuous in nom with respect to z. Moreover, the no ms of theoperators (4.1)-(4.3) at z$=\lambda+i\mathrm{O}$are bounded by$C\lambda^{-1}$

as A$arrow\infty$.

3. Now we are able to check Proposition 4.1. Let us first show that the operators$\langle x\rangle^{p}\langle\xi\rangle^{q}(T_{+}^{(f)})^{*}R(\lambda+i0)T_{-}^{(r)}\langle\xi\rangle^{q}\langle x\rangle^{p}$

are uniformly bounded provided $N$ is large enough. Note that the operators
are bounded by Propositions 3.1 and 3.2 if $(N+1)\rho_{1}\geq\sigma+p-1$ and $N\geq q$ . Thus, itsuffices to use that$\langle x\rangle^{\sigma}T_{\pm}^{(r)}\langle\xi\rangle^{q}\langle x\rangle^{p}$$||\langle x\rangle^{-\sigma}R(\lambda+i0)\langle x\rangle^{-\sigma}||=O(\lambda^{-1/2})$,$\sigma>1/2$,

which follows, for example, from the result of Proposition 4.5 about operator (4.1).
Let us further consider the singular partof . Recall that, according to PropOsition 3.2,
. We need to prove the uniform boundedness of four operators$T_{\pm}^{(s)}$$T_{\pm}$$T_{\pm}^{(s)}\in C_{\pm}^{-1,1}(\Phi_{\pm})$$\langle x\rangle^{p}\langle\xi\rangle^{q}(T_{+}^{(s)})^{*}\mathrm{P}_{-}R(\lambda+i0)\mathrm{P}_{+}T_{-}^{(s)}\langle\xi\rangle^{q}\langle x\rangle^{p}$

,

(4.4)
(4.5)

$\langle x\rangle^{p}\langle\xi\rangle^{q}(T_{+}^{(s)})^{*}\mathrm{P}_{+}R(\lambda+i0)\mathrm{P}_{-}T_{-}^{(s)}\langle\xi\rangle^{q}\langle x\rangle^{p}$

and

.

(4.6)

$\langle x\rangle^{p}\langle\xi\rangle^{q}(T_{+}^{(s)})^{*}\mathrm{r}_{\pm}R(\lambda+i0)\mathrm{p}_{\pm}T_{-}^{(s)}\langle\xi\rangle^{q}\langle x\rangle^{p}$

The operator (4.4) can be factorized into aproduct of three operators$\langle x\rangle^{p}\langle\xi\rangle^{q}(T_{+}^{(s)})^{*}\langle \mathrm{A}\rangle^{-k}$,$\langle \mathrm{A}\rangle^{k}\mathrm{P}_{-}R(\lambda+i0)\mathrm{r}_{+}\langle \mathrm{A}\rangle^{k}$and.

$\langle \mathrm{A}\rangle^{-k}T_{-}^{(s)}\langle\xi\rangle^{q}\langle x\rangle^{p}$

The first and the third factors are bounded for sufficiently large by Proposition 4.3
while the second operator has the form (4.3), and hence it is bounded byby PropO$k$$C\lambda^{-1}$sition 4.5.
The operator (4.5) can be factorized into aproduct of three operators$\langle x\rangle^{p}\langle\xi\rangle^{q}(T_{+}^{(s)})^{*}\mathrm{P}_{+}\langle \mathrm{A}\rangle^{\sigma}$,$\langle \mathrm{A}\rangle^{-\sigma}R(\lambda+i0)\langle \mathrm{A}\rangle^{-\sigma}$and.

$\langle \mathrm{A}\rangle^{\sigma}\mathrm{P}_{-}T_{-}^{(s)}\langle\xi\rangle^{q}\{x\rangle^{p}$

The first and the third factors are bounded for each by Proposition 4.4 while thesecond operator has the form (4.1), and hence it is bounded for any $\sigma>1/2$ by
by
Proposition 4.5.
$\sigma$$C\lambda^{-1}$

Finally, we factorize the operator (4.6) (for the sign $”+”$ , for example) into aproduct of three operators,, $\epsilon>0$ , and. The first factor is bounded for any by Proposition 4.4. The second operator has the form (4.2), and hence it is bounded for any $\sigma>1/2$ by
by
Proposition 4.5. The last factor is bounded by Proposition 4.3 if is sufficiently large.
The cross-terms containing
andcan be considered quite similarly. We need toprove the uniform boundedness of two operators,$\langle x\rangle^{p}\langle\xi\rangle^{q}(T_{+}^{(s)})^{*}\mathrm{P}_{+}\langle \mathrm{A}\rangle^{\sigma}$$\langle \mathrm{A}\rangle^{-\sigma}R(\lambda+i0)\mathrm{P}_{+}\langle \mathrm{A}\rangle^{-1+\sigma-\epsilon}$$\langle \mathrm{A}\rangle^{1-\sigma+\epsilon}T_{-}^{(s)}\langle\xi\rangle^{q}\langle x\rangle^{p}$$\sigma$$C\lambda^{-1}$$\sigma$$T_{+}^{(\tau)}$$T_{-}^{(s)}$$\langle x\rangle^{p}\langle\xi\rangle^{q}(T_{+}^{(r)})^{*}R(\lambda+i0)\mathrm{P}_{\tau}T_{-}^{(s)}\langle\xi\rangle^{q}\langle x\rangle^{p}$

69
where $\tau="+$ ”or $\tau=$ “-,, . First, using Proposition 3.2, for any we can choose $N$
is bounded and hence it suffices to consider thesuch that the operator“-,,
uniformly. If
operators’ then these operators arebounded for any $l>1/2$ according to Proposition 4.4 and the estimate of Proposition 4.5
on the operator (4.1). If $\tau=$ $”+”$ , then according to Proposition 4.3 the operator
is bounded for sufficiently large . So it remains to use that the operator
for $l>k+1$(A)
has the form (4.2), and hence it is bounded byby Proposition 4.5.
This concludes our sketch of the proof of Proposition 4.1 and hence of Theorem 4.2.
$l$$\langle x\rangle^{p}\langle\xi\rangle^{q}(T_{+}^{(r)})^{*}\langle \mathrm{A}\rangle^{l}$$\langle \mathrm{A}\rangle^{-l}R(\lambda+i0)\mathrm{P}_{\tau}T_{-}^{(s)}\langle\xi\rangle^{q}\langle x\rangle^{p}$$\tau=$$k$$\langle \mathrm{A}\rangle^{-k}T_{-}^{(s)}\langle\xi\rangle^{q}\langle x\rangle^{p}$$C\lambda^{-1}$$-lR(\lambda+i0)\mathrm{P}_{+}\langle \mathrm{A}\rangle^{k}$5.

THE SINGULAR PART

1. Let us discuss the precise meaning of the formula (1.12). Recall thatandarbitrary point,is the hyperplane orthogonal to. The coordinates $(z, y)$ inis determined by the condition
by equation (1.11). Set$\omega_{0}\in \mathrm{S}^{d-1}$$\Pi=\Pi_{aJ_{0}}$$\omega_{0}$$\pm\langle\omega, \omega_{0}\rangle>\delta>0$$h_{\pm}(x, \xi)=e^{i\Phi(x,\xi)}\mathrm{b}_{\pm}\pm(x, \xi)$so that$u_{\pm}(x, \xi)=e^{i\langle x,\xi\rangle}h_{\pm}(x, \xi)$is an$\Omega\pm=\Omega\pm(\omega_{0}, \delta)\subset \mathrm{s}^{d-1}$$\mathrm{R}^{d}$

are defined
(5.1)

,

.

Then (1.12) can be rewritten as
,

(5.2)

$s_{0}( \omega, \omega’;\lambda)=(2\pi)^{-d+1}\int_{\Pi}e^{ik\langle y,\omega’-\omega\rangle}\mathrm{a}_{0}(y, \omega, \omega’;\lambda)dy$

where$\omega$,$\omega’\in\Omega\pm$

and$\mathrm{a}_{0}(y,\omega,\omega’;\lambda)=\pm 2^{-1}k^{d-2}(k\langle\omega+\omega’,\omega_{0}\rangle\overline{h_{+}(y,k\omega)}h_{-}(y, k\omega’)$$+ih_{-}(y, k\omega’)\overline{(\partial_{z}h_{+})(y,k\omega)}-i\overline{h_{+}(y,k\omega)}(\partial_{z}h_{-})(y, k\omega’)-2\langle A(y),\omega_{0}\rangle\overline{h_{+}(y,k\omega)}h_{-}(y, k\omega’))(5.3)$

is, actually, regarded as aPDO with amplitude
Formula (5.2) shows that
via its sesquilinear form.
. It is convenient to define the operator
and the orthogonalbyand denote
Indeed, suppose, for example, that.
on the hyperplane which we identify withprojections of and of apointand functions
andWe also identify below points$S_{0}(\lambda)$$S_{0}(\lambda)$$\mathrm{a}_{\mathrm{O}}(y, \omega, \omega’;\lambda)$$\Sigma$$\omega\in\Omega=\Omega_{+}$$\omega$$\in\Omega$$\zeta$$\mathrm{R}^{d-1}$$\Pi$$\omega\in\Omega$$\Omega$$\zeta\in\Sigma$(5.4)$w(\omega)=\tilde{w}(\zeta)$on$\Omega$and X. Set$\tilde{\mathrm{a}}_{0}(y,$$(, \zeta’;\lambda)=(1-|\zeta|^{2})^{-1/2}(1-|\zeta’|^{2})^{-1/2}\mathrm{a}_{0}(y, \omega, \omega’;\lambda)$

Then it follows ffom (5.2) that for arbitrary$w_{j}\in C_{0}^{\infty}(\Omega)$.

, $j=1,2$ ,.

$(S_{0}( \lambda)w_{1}, w_{2})=(2\pi)^{-d+1}\int_{\mathrm{n}}\int_{\mathrm{n}}\int_{\mathrm{n}}e^{ik\langle y,\zeta’-\zeta\rangle}\tilde{\mathrm{a}}_{0}(y, \zeta, \zeta’;\lambda)\tilde{w}_{1}(\zeta’)\overline{\tilde{w}_{2}(\zeta)}d\zeta d\zeta’dy$

(5.5)

70
Since
, the right-hand side of the last equation is well-defined as an
oscillating integral which gives the precise sense to its left-hand side. Of course, we canmake the change of variables $y\mapsto k^{-1}y$ in (5.5) transforming PDO
to the standard
form, but this operation is not really necessary. It follows ffom (5.1) that amplitude (5.3)
contains an oscillating factor
where$\tilde{\mathrm{a}}_{\mathrm{O}}\in S^{0}(\rho, 1-\rho)$$S_{0}(\lambda)$$\exp(i_{-}^{-}-)$—(y,and hence the operator$\omega,\omega’;k)$$S_{0}(\lambda)$$=\Phi_{-}(y, k\omega’)-\Phi_{+}(y, k\omega)$

,

(5.6)

is bounded according to Proposition 3.1.

2. It follows from Theorem 4.2 that the operator (1.17) contains all power termsof the high-energy expansion of the SM as well as of its diagonal singularity. However,
the obvious drawback of the expression (1.17) is that it depends on the cut-0ff functions. Our final goal is to show that, up to negligible terms, it can be transformed to theinvariant expression (1.12).
We proceed from relation (3.6) where. Recall that
andare PDOdefined by formulas (3.3) and (3.4), (3.5), respectively. Therefore for all , $f_{2}\in S$$\zeta_{\pm}$$\mathrm{A}=J_{+}^{*}T_{-}$$T_{-}$$J_{+}$$f_{1}$$(T_{-}f_{1}, J_{+}f_{2})=(2 \pi)^{-d}\int_{\mathrm{R}^{d}}(\int_{\mathrm{n}^{d}}\int_{\mathrm{R}^{d}}e^{:(xk’-\xi\}}\mathrm{a}(x,\xi,\xi’)\hat{f}_{1}(\xi’)\overline{\hat{f}_{2}(\xi)}d\xi d\xi’)dx$

,

(5.7)

where
(5.8)

$\mathrm{a}(x,\xi, \xi’)=\overline{j_{+}(x,\xi)}t_{-}(x,\xi’)$and ,are the symbols of the operators ,, respectively. According to Propositions 2.2, 2.4 and 3.2, the amplitude$1-$
belongs to the Hormander class. To obtain aconvenient representation for (5.7), we have to change the order of integrations over andin (5.7) and then calculate the integral over . Below we do
not go into details of standard manipulations with oscillating integrals. Note only that,strictly speaking, we have to introduce into (5.7) afunction,
such that$\varphi(0)=1$ , and pass to the limit $6arrow 0$ at the very end of ourcalculations. Denote$j_{+}$$t_{-}$$J_{+}$$T_{-}$$\mathrm{a}(x,\xi,\xi’)$$S^{-1}(\rho,$$\rho)$$x$$\xi,\xi’$$x$$\varphi\in C_{0}^{\infty}(\mathrm{R}^{d})$$\varphi(\epsilon x)$(5.9)$G( \xi,\xi’)=\int_{\mathrm{R}^{d}}e^{:\{xi’-\xi)}.\mathrm{a}(x,\xi,\xi’)dx$

be integral operator with kernel $G(\xi,\xi’)$ . Then, at least formaly, $G=$
. We set (, then $\zeta_{+}(x,\xi)=\zeta(x, -\xi)$ . It follows from (3.3), (3.5)(5.8),(5.9) thatandand let$G$$(2\pi)^{d}\mathcal{F}J_{+}^{*}T_{-}\mathcal{F}^{*}$$=\zeta_{-}$.

(5.10)$G( \xi,\xi’)=\int_{\mathrm{R}^{d}}\overline{u_{+}(x,\xi)}\zeta(x, -\xi)g_{-}(x,\xi’)dx$

Standard arguments show that, off the diagonal, $G(\xi,\xi’)$ is asmooth function, andit rapidly tends to zero as
and. Applying (3.6) to functions
andwith disjoint supports, it is easy to show that off the diagonal
the kernelof the operator
satisfies the relation (cf. (3.7))$|\xi|arrow\infty$$|\xi’|arrow\infty$$w_{1}$$w_{2}$$s_{1}(\omega,\omega’, \lambda)$$\omega$$=\omega’$$S_{1}(\lambda)$$s_{1}(\omega,\omega’;\lambda)=-\pi ik^{d-2}G(k\omega, k\omega’)$

Combining these results with Theorem 4.2, we obtain,$\omega$$\neq\omega’$.

(5.11)

71,
Theorem 5.1 Let assumption (1.7) hold, and let
for some open sets,such that dist $(\Omega, \Omega’)>0$ . Then for any p and q the kernelof the.
byasAboundedSM belongs to the space-norm
isand its$\omega\in\Omega$$\Omega$$\omega’\in\Omega’$$\Omega’\subset \mathrm{S}^{d-1}$$s(\omega, \omega’, \lambda)$$C\lambda^{-q}$$C^{p}$$C^{p}(\Omega\cross\Omega’)$$arrow\infty$

3. Our study of the function (5.10) in aneighborhood of the diagonal $\xi=\xi’$ relies on, $j=1,2,3,4$ ,integration by parts. Let us plug (3.4) into (5.10) and denote by(3.4):
sideofright-handintheintegralsthecorresponding to the four functions$G_{j}(\xi, \xi’)$,$G_{1}(\xi, \xi’)$$=$$\int_{\mathrm{R}^{d}}\overline{u_{+}(x,\xi)}\zeta(x, -\xi)q_{-}(x, \xi’)\zeta(x, \xi’)dx$$G_{2}(\xi, \xi’)$

$=$

-2

$G_{3}(\xi,\xi’)$$=$$- \int_{\mathrm{R}^{d}}\overline{u_{+}(x,\xi)}\zeta(x, -\xi)u_{-}(x, \xi’)\Delta\zeta(x, \xi’)dx$$G_{4}(\xi, \xi’)$$=$$2i \int_{\mathrm{R}^{d}}\overline{u_{+}(x,\xi)}\zeta(x, -\xi)u_{-}(x$,$\int_{\mathrm{R}^{d}}\overline{u_{+}(x,\xi)}\zeta(x, -\xi)\langle\nabla u_{-}(x,\xi’), \nabla\zeta(x,\xi’)\rangle dx$, ’$\langle$,$)\langle A(x), \nabla\zeta(x,\xi’)\rangle dx$.

. By virtue of Theorem 2.6,
Let us consider first the function
whereand the function
the function
satisfies estimates (2.17) for all ,$r_{-}(x, \xi’)\zeta(x,\xi’)$ satisfies estimates (2.16) for all. Hence the integrand in,is estimated by $C|\xi|^{-N}(1+|x|)^{-1-\rho_{1}(N+1)}$ , where $N$ can be chosen arbitrary large. Using
also the estimates on derivatives of these functions and estimates (2.13) on the phase, we see thatfunctionsis asmooth function of , rapidly decreasing as$G_{1}$$q_{-}=e^{i\Theta_{-}}r_{-}$$\overline{\mathrm{b}_{+}(x,\xi)}\zeta(x, -\xi)$

$x$

$x$

$G_{1}(\xi, \xi’)$$\Phi_{\pm}$$|\xi|=|\xi’|arrow\infty$$\xi\in \mathrm{R}^{d}$$G_{1}(\xi,\xi’)$$\xi’\in \mathrm{R}^{d}$$\xi$$\xi’$.

where, for exLet andbelong to some conical neighborhood of apoint$\zeta(x,-\xi)(\nabla\zeta)(x,\xi’)=(\nabla\zeta)(x,\xi’)$. Thenample,
so that the function $\zeta(x, -\xi)$in the integrals, $j=2,3,4$ , can be omitted. All these integrals will be transby parts, we find thatformed by integration by parts. Integrating in$\omega$$\omega_{1}\in \mathrm{S}^{d-1}$$\omega’$$\langle\omega_{1},\omega_{0}\rangle>0$$G_{j}(\xi, \xi’)$$G_{3}(\xi, \xi’)$$G_{2}(\xi,\xi’)+G_{3}(\xi,\xi’)=$$+ \int_{\mathrm{R}^{d}}\langle u_{-}(x, \xi’)\overline{(\nabla u_{+})(x,\xi)}-\overline{u_{+}.(x,\xi)}(\nabla u_{-})(x, \xi’), \nabla\zeta(x,\xi’)\rangle dx$.

(5.12)

are actually taken, the integrals (5.12) as well as
Due to the function
over the half-space $z\geq 0$ only. Therefore integrating once more by parts and taking into
account the equality $\zeta(y, \xi’)=1$ , we obtain that
$G_{4}(\xi,\xi’)$$\nabla\zeta(x, \xi’)$$G_{2}(\xi,\xi’)+G_{3}(\xi,\xi’)=$$+ \int_{z\geq 0}(\overline{u_{+}(x,\xi)}(\Delta u_{-})(x, \xi’)-u_{-}(x, \xi’)\overline{(\Delta u_{+})(x,\xi)})\zeta(x,\xi’)dx$$+ \int_{\mathrm{n}}(\overline{u_{+}(y,\xi)}(\partial_{z}u_{-})(y, \xi’)-u_{-}(y,\xi’)\overline{(\partial_{z}u_{+})(y,\xi)})dy$(5.13)

and$G_{4}( \xi,\xi’)=-2i\int_{z\geq 0}\mathrm{d}\mathrm{i}\mathrm{v}(A(x)\overline{u_{+}(x,\xi)}u_{-}(x, \xi’))\zeta(x,\xi’)dx$$-2i \int_{\mathrm{n}}\langle A(y),\omega_{0}\rangle\overline{u_{+}(y,\xi)}u_{-}(y,\xi’)dy$

It is now convenient to formulate an intermediary result.

(5.14)

72
Proposition 5.2 The function (5.10) is the sum$G=G_{1}+G_{2}+G_{3}+G_{4}$Here$G_{1}(\xi, \xi’)$functionsrapidly decreasing as $|\xi|=|\xi’|arrow\infty$ . The
satisfy equalities (5.13) and (5.14), respectively.

is a smooth function of$G_{2}+G_{3}$and$G_{4}$.

$\xi,\xi’$

4. In the following we need to calculate the operators
for two special classes ofintegral operators
acting on functions of (6 . For the operators from thefirst class the passage to the limit (3.6) is quite direct (cf. (3.7)).
$\mathrm{A}^{\mathrm{b}}(\lambda)$$G=\mathcal{F}\mathrm{A}P$$\mathrm{R}^{d}$

Proposition 5.3 Let an operator $G$ be defined by its kernel,$G( \xi,\xi’)=\int_{\mathrm{n}}e^{:\langle y,\xi’-\xi)}\mathrm{a}(y, \xi, \xi’)dy$

where$\lambda>0$

and $\rho>0$ , $\delta<1$ . Then the $0\mu$ rator
and is the integral operator on the unit sphere with kernel$\mathrm{a}\in S^{p}(\rho, \delta)$

for some$p$$g( \omega,\omega’;\lambda)=2^{-1}k^{d-2}\int_{\mathrm{n}}e^{:k\{y\mu’-\mathrm{I}v)}\mathrm{a}(y, !, k\omega’)dy$,$A^{\mathfrak{d}}(\lambda)$$\omega,\omega’\in\Omega_{\pm}$exists for all.

Kernels of the operators from the second class are defined in terms of integrals over a
half-space.

Proposition 5.4 Let an operator G have kernel$G( \xi,\xi’)=(|\xi|^{2}-|\xi’|^{2})\int_{z\geq 0}e^{:\{x,\xi’-\xi)}\mathrm{a}(x,\xi,\xi’)dx$

where$\mathrm{a}\in S^{p}(\rho, \delta)$

for some$\mathrm{a}(x,\xi$

for some$c_{0}\in$(0, 1). Then$p$,

(5.15)

and $\rho>0$ , $\delta<1$ . Assume moreover that

, $(’)=0$$\mathrm{A}^{\}}(\lambda)=0$

if$\langle\xi+\xi’,x\rangle\geq c_{0}|\xi+\xi’||x|$

for all

The proof relies on condition (5.16). Let$\lambda>0$(5.16).

$\mathrm{A}_{1}=\mathcal{F}^{\vee}G_{1}\mathcal{F}$

where

.

$G_{1}( \xi,\xi’)=\int_{z\geq 0}e^{:\{x,\xi’-\xi\}}\mathrm{a}(x,\xi,\xi’)dx$

Then the operator
the factor-defined(cf. [15, 25]) due to (5.16). Taking into accountis.
in (5.15), it is easy to show that$\mathrm{A}_{1}^{\mathrm{b}}(\lambda)$$|\xi|^{2}-|\xi’|^{2}$$\mathrm{w}\mathrm{e}\mathrm{U}$$\mathrm{A}^{\mathrm{b}}(\lambda)=0$.
5. Now we are in aposition to derive formula (1.12) for the singular part of the
To that end, we have to calculate the limit in the right-hand side of (3.6) forand show that the expression obtained coincides, up to negligible terms, with the form
. Let us proceed from Proposition 5.2.
tois given by the expression $-\pi ik^{d-2}$
According to (3.7) the contribution ofand rapidly decays as $karrow\infty$ . Hence
which is asmooth function of ,this term can be neglected$\mathrm{S}\mathrm{M}$$\mathrm{A}=J_{+}^{*}T_{-}$$-(2\pi i)^{-1}(S_{0}(\lambda)w_{1},w_{2})$$G_{1}$$\mathrm{x}G_{1}(k\omega, k\omega’)$$S_{1}(\lambda)$$\omega$$\omega’$73
Let us further consider the integrals (5.13) and (5.14) over . By virtue of Proposiequals its value at $\xi=k\omega$ ,tion 5.3, the contribution of each integral to the kernel of$-\piik^{d-2}(2\pi)^{-d}$ .
The sum of$\xi’=k\omega’$ times (compare with (5.11)) the numerical factor
this expressions coincides with (1.12).
It remains to show that the sum of the integrals over the half-s ace $z\geq 0$ in (5.13)
and (5.14) is negligible. It follows from relation (1.10) that$\Pi$

$S_{1}(\lambda)$$\overline{u_{+}(x,\xi)}(\Delta u_{-})(x,\xi’)-u_{-}(x,\xi’)\overline{(\Delta u_{+})(x,\xi)}-2i\mathrm{d}\mathrm{i}\mathrm{v}(A(x)\overline{u_{+}(x,\xi)}u_{-}(x,\xi’))$$=(\overline{q_{+}(x,\xi)}u_{-}(x,\xi’)-q-(x,\xi’)\overline{u_{+}(x,\xi)})+(|\xi|^{2}-|\xi’|^{2})\overline{u_{+}(x,\xi)}u_{-}(x,\xi’)$.

To consider the integral(5.17),$\int_{z\geq 0}e^{i\Theta_{-}(x,\xi’)-i\Theta_{+}(x,\xi)}(\overline{r_{+}(x,\xi)}\mathrm{b}_{-}(x, \xi’)-r_{-}(x, \xi’)\overline{\mathrm{b}_{+}(x,\xi)})\zeta(x, \xi’)dx$

we use again that, by Proposition 2.4 and Corollary 2.5, the functions.
satisfy estimates (2.16) and (2.17), respectively, for all ,and$z\geq 0$
ace
half-s
the
in
true
holds
and
The same result for the functions. By Corollary 2.3, the function
which does not contain the “bad” direction$\Phi_{-}(x, \xi’)-\Phi_{+}(x, \xi)$ satisfies estimates (2.13) for all $z\geq 0$ off aconical neighborhood of
where $\zeta(x, \xi’)=0$ . Therefore the integral (5.17) is asmooth function
the direction,
rapidly decreasing as $|\xi|=|\xi’|arrow\infty$ . Hence, similarly to the functionof.
this integral does not contribute to
Let us, finally, consider the kernel$r_{-}(x, \xi’)\zeta(x, \xi’)$$x$$\mathrm{b}_{-}(x, \xi’)\zeta(x, \xi’)$$\xi’\in \mathrm{R}^{d}$$r_{+}(x, \xi)$$\mathrm{b}_{+}(x, \xi)$$\hat{x}=-\hat{\xi}$$\hat{x}=\hat{\xi}’$$G_{1}(\xi, \xi’)$$\xi,\xi’$$S_{0}(\lambda)$$G_{0}( \xi, \xi’)=(|\xi|^{2}-|\xi’|^{2})\int_{z\geq 0}e^{i\langle x,\xi’-\xi\rangle}\overline{h_{+}(x,\xi)}h_{-}(x, \xi’)\zeta(x, \xi’)dx$,

are defined by formula (5.1). Due to the factor
where the functions
satisfies the conditions of Proposition 5.4 and hence
function.
for all
Now we can formulate our main result on the asymptotics of the kernel.
the$h_{\pm}(x, \xi)$$\zeta(x, \xi’)$, the$(\mathcal{F}^{*}G_{0}\mathcal{F})^{\mathrm{b}}(\lambda)=0$$G_{0}(\xi, \xi’)$$\lambda>0$$s(\omega,\omega’;\lambda)$

of$\mathrm{S}\mathrm{M}$Theorem 5.5 Let assumption (1.7) hold, let , be arbitrar$ry$ numbers and $N=N(p, q)$
be constructed in Proposiand
be sufficiently large. Let functions
be defined by formula (1.9). Define, fortions 2.2 and 2.4, respectively, and let
by for mula (1.12). Then the remainder (1.13) belongs, the kernel,.
as-norm of this kernel isand theto the class$p$$_{\pm}^{(N_{0})}(x, \xi)$$q$$\mathrm{b}_{\pm}^{(N)}(x, \xi)$$u_{\pm}^{(N)}(x, \xi)$$\omega$$s_{0}^{(N)}(\omega, \omega’;\lambda)$$\omega’\in\Omega_{\pm}$$C^{p}(\Omega\cross\Omega)$$C^{p}$$O(\lambda^{-q})$$\lambdaarrow\infty$This result gives simultaneously the high-energy and smoothness expansion of the. As was already mentioned, we actually formulate the result in terms
kernel of therelated to the kernel of the SM by formulaof the corresponding amplitude(5.2). Indeed, it follows from (5.1), (5.3) and (5.6) that$\mathrm{S}\mathrm{M}$$\mathrm{a}_{0}(y, \omega, \omega’;\lambda)$$\mathrm{a}_{0}(y, \omega,\omega’;\lambda)=\pm 2^{-1}k^{d-1}\exp(i_{-}^{-}-(y, \omega, \omega’;k))\sum_{n=0}^{N}(2ik)^{-n}\sigma_{n}(y, \omega, \omega’)$

,

74

—(y,

$\omega,\omega’;k)$$= \sum_{n=0}^{N_{\mathrm{O}}}(2k)^{-n}\theta_{n}(y,\omega,\omega’)$,$\theta_{n}(y,\omega,\omega’)=\phi_{n}^{(-)}(y,\omega’)-\phi_{n}^{(+)}(y,\omega)$

and the functions
a constructed in Proposition 2.2. Note thatandfor $n\geq 1$ . The coefficientsare expressed in terms of functions
andconstructed in Proposition 2.4. It is easy to see thatfor $n\geq 0$ . In
particular, 50.
$\phi_{n}^{(\pm)}$$\theta_{n}\in S^{1-n\rho}$$\theta_{0}\in S^{1-\rho_{\alpha}}$$\mathrm{e}$$\sigma_{n}(y,\omega,\omega’)$$b_{n}^{(\pm)}$$\phi_{n}^{(\pm)}$$\sigma_{n}\in S^{-n\rho 1}$$(\mathrm{A})\in C^{0}(_{-}^{-}-)$REFERENCES[1] S. Agmon, Some new results in spectral and scattering theory of differential operators
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