Equidistribution of Phase Shifts in Obstacle Scattering

TL;DR

This paper proves that phase shifts in obstacle scattering become uniformly distributed on the unit circle at high frequencies for convex obstacles, providing new insights into spectral asymptotics and scattering theory.

Contribution

It establishes the equidistribution of scattering matrix eigenvalues for convex obstacles and offers an alternative proof of the total scattering phase asymptotics.

Findings

Eigenvalues of the scattering matrix equidistribute on the unit circle as frequency increases.

Number of eigenvalues in sectors scales with $k^{d-1}$ and the sector size.

Provides an alternative proof of the two-term asymptotic expansion of the total scattering phase.

Abstract

For scattering off a smooth, strictly convex obstacle $\Omega \subset \mathbb{R}^d$ with positive curvature, we show that the eigenvalues of the scattering matrix -- the phase shifts -- equidistribute on the unit circle as the frequency $k \to \infty$ at a rate proportional to $k^{d - 1}$, under a standard condition on the set of closed orbits of the billiard map in the interior. Indeed, in any sector $S \subset \mathbb{S}^1$ not containing $1$, there are $c_d |S| \mathrm{Vol}(\partial \Omega)\ k^{d - 1} + o(k^{d-1})$ eigenvalues for $k$ large, where $c_d$ is a constant depending only on the dimension. Using this result, the two term asymptotic expansion for the counting function of Dirichlet eigenvalues, and a spectral-duality result of Eckmann-Pillet, we then give an alternative proof of the two term asymptotic of the total scattering phase due to Majda-Ralston.

Figures

Equations

$$(\Delta +k^2)u=0u{|}_{\partial \Omega }=0$$

$$u(x)=|x|^{-(d-1)/2}\big{(}e^{-ik|x|}\phi_{in}(-\widehat{x})+e^{ik|x|}\phi_{out}(\widehat{x})\big{)}+O_{|x|\rightarrow\infty}(|x|^{-(d+1)/2}),$$

$$S(k)({\varphi }_{in}):=e^{i\pi (d-1)/2}{\varphi }_{out}.$$

$$N_k({\varphi }_0,{\varphi }_1,\Omega )=N_k({\varphi }_0,{\varphi }_1):=\#\{e^{i{\beta }_{k,n}}\in \sigma (S(k)):{\varphi }_0<{\beta }_{k,n}<{\varphi }_1,\mathrm{mod}2\pi \}.$$

$$N_{k}(\phi_{0},\phi_{1})=\frac{\omega_{d-1}}{(2\pi)^{d-1}}\Big{(}\frac{\phi_{1}-\phi_{0}}{2\pi}\Big{)}\mathrm{Vol}(\partial\Omega)k^{d-1}+o(k^{d-1}).$$

$$\langle\mu_{k},f\rangle=\Big{(}\frac{2\pi}{k}\Big{)}^{d-1}\sum_{\sigma(S(k))}f(e^{i\beta_{k,n}}).$$

$$\underset{k\to \infty }{\lim }⟨{\mu }_k,f⟩=\frac{\mathrm{Vol}(\partial \Omega ){\omega }_{d-1}}{2\pi }{\int }_0^{2\pi }f(e^{i\theta })d\theta$$

$$s(k)=i\log \det S(k).$$

$$s(k)=\frac{{\omega }_d}{(2\pi {)}^{d-1}}\mathrm{Vol}(\Omega )k^d+\frac{{\omega }_{d-1}}{4(2\pi {)}^{d-2}}\mathrm{Vol}(\partial \Omega )k^{d-1}+o(k^{d-1}).$$

$$N_D({\lambda }_0):=\#\{0<\lambda <{\lambda }_0:\exists \varphi \in L^2(\Omega ),\varphi {|}_{\partial \Omega }=0,\Delta \varphi =-{\lambda }^2\varphi ,\varphi \ne 0\}.$$

$$\nu=\left\{\begin{array}[]{cc}c_{1}\theta^{-(\gamma+1)}&\mbox{ for }\theta>0\\ c_{2}|\theta|^{-(\gamma+1)}&\mbox{ for }\theta<0\end{array}\right.,$$

$$\mathcal{I}:=\{(\omega ,\eta )\in T^{\ast }{\mathbb{S}}^{d-1};L_{(\omega ,\eta )}\cap \partial \Omega \text{ contains two points}\}.$$

$$y_{\pm }(\omega ,\eta )$$

$${\omega }^{\prime }(\omega ,\eta )$$

$${\eta }^{\prime }(\omega ,\eta )$$

$$\kappa (\omega ,\eta )=({\omega }^{\prime },{\eta }^{\prime }).$$

$$\mathrm{Vol}(\mathcal{I})=\mathrm{Vol}(\partial \Omega ){\omega }_{d-1}.$$

$$\mathcal{I}=T^{\ast }{\mathbb{S}}^{d-1}\backslash {\mathcal{P}}_1,$$

$${\mathcal{P}}_p^{\prime }:={\mathcal{P}}_p\setminus {\mathcal{P}}_1,$$

$$\kappa_{int}\big{(}(y_{-}(\omega,\eta),-\omega\big{)}=\big{(}y_{-}(\kappa(\omega,\eta),\omega^{\prime}(\kappa(\omega,\eta)\big{)}.$$

$$\forall p\in \mathbb{Z}\backslash \{0\},\mathrm{Vol}({\mathcal{P}}_p^{\prime })=0.$$

$$\forall p\in \mathbb{Z}\backslash \{0\},\mu ({\mathcal{P}}_p^{int})=0,$$

$$A(k):=S(k)-\mathrm{Id}.$$

$$\begin{array}{rl}(\Delta +k^2)v& =0\\ v{|}_{\partial \Omega }& =-e^{ix\cdot \xi },\end{array}$$

$$v(|x|\omega ;\xi ,k)=|x{|}^{-(d-1)/2}{e}^{ik|x|}a(\omega ,\xi ,k)+O(|x{|}^{-(d+1)/2}).$$

$$A(k)f(\omega )={\int }_{{\mathbb{S}}^{d-1}}a(\omega ,\theta ,k)f(\theta )dVo{l}_{{\mathbb{S}}^{d-1}}(\theta ),$$

$$a(\omega ,\theta ,k)$$

$$+\frac{1}{2i}{k}^{d-2}(2\pi {)}^{1-d}{\int }_{\partial \Omega }{e}^{-ik\theta \cdot y}\frac{\partial }{\partial \nu }v(y,-k\omega )dVo{l}_{\partial \Omega }(y).$$

$$\frac{\partial v(x,-k\omega )}{\partial \nu }=K(\omega ,x,k)e^{ikx\cdot \omega }.$$

$$K(\omega ,x,k)=-ik|{\nu }_x\cdot \omega |+kE(\omega ,x,k),$$

$$E\in k^{-1/3}{S}_{1/3}({\mathbb{S}}^{d-1}\times \partial \Omega ).$$

$$||E(\cdot ,\cdot ,k)|{|}_{C^0}\le C{k}^{-1/3}.$$

Full text

1. Introduction

Let $\Omega\subset\mathbb{R}^{d}$ denote a strictly convex domain whose boundary $\partial\Omega$ is smooth and has positive sectional curvature. We shall write $\Omega^{c}:=\mathbb{R}^{d}\backslash\Omega$. It is well-known (see for instance [Mel95, §5] or [DZ, §4.4]) that for any $k>0$ and any $\phi_{in}\in C^{\infty}(\mathbb{S}^{d-1})$, there is a unique solution $u\in C^{\infty}(\overline{\Omega^{c}})$ to the Dirichlet problem

[TABLE]

such that

[TABLE]

where we write $\widehat{x}=\frac{x}{|x|}\in\mathbb{S}^{d-1}$ and $\Delta=\sum_{i=1}^{d}\partial_{x_{i}}^{2}$. In particular $\phi_{out}$ is determined by $\phi_{in}$ and we define the scattering matrix $S(k)$, which depends on $k$ and $\Omega$, by

[TABLE]

$S(k)$ extends to a unitary operator acting on $L^{2}(\mathbb{S}^{d-1})$ with the property that $S(k)-\mathrm{Id}$ is trace class [Tay11, RS79]. Therefore, for any $k>0$, $S(k)$ has purely discrete spectrum, accumulating only at 1, which we denote by $\sigma(S(k)):=\{e^{i\beta_{k,n}}\}$. Our aim in this paper is to study the asymptotic distribution of the $e^{i\beta_{k,n}}$ as $k\rightarrow\infty$.

Our main result is an estimate for the number of phase shifts in a sector $S\subset\mathbb{S}^{1}\setminus\{1\}$ as $k\to\infty$. Define the counting function

[TABLE]

where the eigenvalues are counted according to multiplicity. Letting $\omega_{d-1}=|B^{d-1}|$ where $B^{d-1}$ is the unit ball in $\mathbb{R}^{d-1}$, we will prove

[TABLE]

In particular, the phase shifts accumulate in each sector $S$ at a rate proportional to $k^{d-1}$ as $k\to\infty$ times $\mathrm{Vol}(\partial\Omega)|S|$. The estimate in (1.2) follows from Theorem 1.1, see Section 5.

To study the asymptotic distribution of the phase shifts, consider the measure $\mu_{k}$ on the circle $\mathbb{S}^{1}$, defined for continuous functions $f\colon\mathbb{S}^{1}\longrightarrow\mathbb{C}$ by

[TABLE]

Note that $\langle\mu_{k},f\rangle$ is finite if $1\not\in\operatorname{supp}f$. The following theorem describes the behavior $\mu_{k}$ as $k\rightarrow\infty$, provided (2.6) holds, which is a standard assumption on the volume of the periodic points of the inside billiard map. Note that this assumption holds if our smooth convex obstacle, is generic (see the discussion at the end of Section 2).

Theorem 1.1**.**

Let $\Omega\subset\mathbb{R}^{d}$ be a smooth strictly convex open set with positive sectional curvature, such that (2.6) holds. Then for any $f\colon\mathbb{S}^{1}\longrightarrow\mathbb{C}$ with $\operatorname{supp}f\cap\left\{1\right\}=\varnothing$, we have

[TABLE]

Remark 1.2**.**

The factor in front of the integral in (1.4) arises as the volume of the ‘interacting region’ in phase space of incoming rays from the sphere at infinity that make contact with the obstacle. See Section 2 for further description of the classical dynamics. In [GRHZ15], in which the first author and collaborators studied the same problem for semiclassical potential scattering, they defined a measure $\mu_{h}$, depending on a semiclassical parameter $h\to 0$, analogously to the measure in (1.3) except they included the volume of the interacting region. Here we prefer not to, so that the dependence on the interacting region appears explicitly in the limit measure.

As an application of the equidistribution of the measure $\mu_{k}$, we will give an alternative proof of the following result of Majda-Ralston, generalized by Melrose and then by Robert, regarding the asymptotic development of the total scattering phase

[TABLE]

The scattering phase $s(k)$ can be defined in a natural way so that $s(k)\in C^{\infty}((0,\infty))$.

Theorem 1.3** ([MR78, Mel88, Rob96]).**

Let $\Omega$ be a smoothly bounded, strictly convex obstacle whose set of periodic billiard trajectories has measure zero. Then

[TABLE]

In fact, Theorem [Mel88, Rob96] holds for all smoothly bounded, compact domains satisfying the stated assumption on the periodic trajectories.

As we describe in Section 5, the novelty in our proof comes from its use of the explicit relationship between the counting function for the Dirichlet eigenvalues,

[TABLE]

and the scattering phase which arises from the spectral duality result of Eckmann-Pillet [EP95]. Indeed, note that the leading order term in (1.6) is $2\pi$ times the leading order term in Weyl’s law [Ivr80], which is to be expected since, as explained in Section 5, ‘inside-outside’ duality says that a phase shift makes a complete rotation of the unit circle for each Dirichlet eigenvalue of $\Omega$.

Relation to other works

Since the pioneering works of Birman, Sobolev, and Yafaev (see for example [SY85, BY84]), there has been a wealth of literature on the asymptotic behavior of the scattering matrix at high energy, in particular about the distribution of phase shifts. In semi-classical potential scattering, an analogous result for compactly supported potentials was proven by the first author, Hassell, and Zelditch in [GRHZ15] for non-trapping potentials, and was generalized to trapping potentials by the second author in [Ing16a]. See [GRHZ15] for a complete literature review of phase shift asymptotics for potential scattering. The behaviour of the phase shifts in the semi-classical limit has been studied in various settings: for magnetic potentials ([BP12]), for scattering by radially symmetric potentials, in [DGRHH13], near resonant energies in [NP14]…

The idea of using trace formulae to analyze the asymptotics of the spectra comes from [Zel92, Zel97], and was the starting point of [GRHZ15], [Ing16a] and of the present paper. The main tool we use here is the Kirchhoff approximation, which was proven in its optimal form in [MT85]. Finally, our proof is simplified by describing the micro-local properties of the scattering matrix in terms of its action on Gaussian states, an approach which was introduced in [Ing16a] for potential scattering.

There do exist perturbations of the free Hamiltonian for which the phase shifts do not equidistribute. Indeed, for Schrödinger operators of the form $\Delta+V$ where $V\sim v_{0}(\widehat{x})/|x|^{\alpha}$, $\alpha>d$, the first author and Hassell showed in [GRH15] that an appropriatly rescaled spectral measure for $S_{h}$ converges to the pushforward via the map $\mathbb{R}\longrightarrow\mathbb{R}/2\pi\mathbb{Z}=\mathbb{S}^{1}$ of a homogeneous measure

[TABLE]

on $\mathbb{R}$, where $\gamma=(d-1)/(\alpha-1)$ and the $c_{1},c_{2}$ are determied by $v_{0}$. It would be interesting to know if there are circumstances under which equidistribution fails in the setting of obstacle scattering.

Organisation of the paper

In Section 2, we will recall a few facts about the classical scattering dynamics, and its links with the interior billiard dynamics. In Section 3, we will recall the main tools we use to prove Theorem 1.1. In Section 4, we prove Theorem 1.1 in the special case when $f$ is a polynomial vanishing at one. Finally, we prove Theorems 1.1 and 1.3 in Section 5. The appendix contain rather elementary facts of semiclassical analysis, and a proof of a resolution of identity formula on the sphere.

Acknowledgements

J.G.R. acknowledges the support of the Australian Research Council through Discovery Grant DP180100589. M.I. was funded by the LabEx IRMIA, and partially supported by the Agence Nationale de la Recherche project GeRaSic (ANR-13-BS01-0007-01). Both authors wish to thank the Australian Mathematical Sciences Institute and the Mathematical Sciences Institute at the Australian National University for their partial funding of the workshop “Microlocal Analysis and its Applications in Spectral Theory, Dynamical Systems, Inverse Problems, and PDE” at which part this project was completed.

2. Classical scattering dynamics and interior dynamics

Let $\omega\in\mathbb{S}^{d-1}$ and $\eta\in\omega^{\perp}\subset\mathbb{R}^{d}$. We will always identify $(\omega,\eta)$ with a point in $T^{*}\mathbb{S}^{d-1}$. Consider the line $L_{(\omega,\eta)}:=\{t\omega+\eta,t\in\mathbb{R}\}$. By strict convexity of $\partial\Omega$, it intersects $\partial\Omega$ in zero, one or two points. We define the interaction region,

[TABLE]

If $(\omega,\eta)\in\mathcal{I}$, then there exists $t_{-}<t_{+}$ such that $t_{\pm}\omega+\eta\in\partial\Omega$. We then set (see Figure 1)

[TABLE]

where $\nu_{y}$ is the outward pointing normal vector at the point $y\in\partial\Omega$. We then set

[TABLE]

If $(\omega,\eta)\notin\mathcal{I}$, we shall set $\kappa(\omega,\eta)=(\omega,\eta)$. The map $\kappa$ may then be seen as a $C^{0}$ map $\kappa:T^{*}\mathbb{S}^{d-1}\rightarrow T^{*}\mathbb{S}^{d-1}$, which is smooth (and even symplectic) away from the glancing set $\partial\Omega$.

Using Cauchy’s surface area formula, it is straightforward that

[TABLE]

For $p\in\mathbb{Z}\backslash\{0\}$, we will denote by $\mathcal{P}_{p}\subset T^{*}\mathbb{S}^{d-1}$ the set of fixed points of $\kappa^{p}$. Note that we then have

[TABLE]

and that $\partial\mathcal{P}_{1}=\partial\mathcal{I}$ is exactly the ‘glancing set’, i.e. the set of $(\omega,\eta)$ such that $L_{(\omega,\eta)}\cap\partial\Omega$ consists of a single point. We define

[TABLE]

the set of non-trivial glancing periodic points with period $p$, also an invariant subset.

The sets $\mathcal{P}^{\prime}_{p}$ will play a central role in our proof, and can be better understood in terms of the periodic points of the interior billiard map, as follows. Consider the set $\mathcal{O}:=\{(y,\xi)\in S^{*}\partial\Omega;~{}\xi\cdot\nu_{y}<0\}$. If $(y,\xi)\in\mathcal{O}$, there will be a unique $t>0$ such that $y+t\xi\in\partial\Omega$. We shall then write $y^{\prime}(y,\xi)=y+t\xi$, and $\xi^{\prime}(y,\xi)=\xi-2(\xi\cdot\nu_{y^{\prime}})\nu_{y^{\prime}}$. We have $(x^{\prime},\xi^{\prime})\in\mathcal{O}$, and we may define $\kappa_{int}:\mathcal{O}\rightarrow\mathcal{O}$ by $\kappa_{int}(x,\xi)=(x^{\prime},\xi^{\prime})$. The map $\kappa_{int}$, and we shall denote by $\mathcal{P}_{p}^{int}$ the set of periodic points of period $p$ of $\kappa_{int}$.

The following elementary lemma makes explicit the link between $\kappa$ and $\kappa_{int}$, as can be seen on Figure 2.

Lemma 2.1**.**

Let $(\omega,\eta)\in T^{*}\mathbb{S}^{d-1}\backslash\mathcal{P}_{1}$. We then have

[TABLE]

As a consequence of this lemma, we see that $\mathcal{P}^{\prime}_{p}$ is homeomorphic to $\mathcal{P}_{p}^{int}$.

The volume of the set of fixed points

Let us denote by $\mathrm{Vol}$ the (symplectic) volume on $T^{*}\mathbb{S}^{d-1}$. We will always assume that we have

[TABLE]

This condition may of course be rephrased in terms of the dynamics of $\kappa_{int}$ on $\mathcal{O}$. If $\mu_{\mathcal{O}}$ is any Riemannian volume and $d_{\mathcal{O}}$ is any Riemannian distance on the manifold $\mathcal{O}$, Equation (2.5) is equivalent to

[TABLE]

Condition (2.6) is conjectured to hold for all domains $\Omega\subset\mathbb{R}^{d}$, not necessarily convex. This conjecture, known as Ivrii’s conjecture, has implications in terms of remainders for the Weyl’s law for the eigenvalues of the Laplacian (see [Ivr80]). In the generic case, it was shown in [PS88] that $\mathcal{P}^{int}_{p}$ is finite for all $p\in\mathbb{Z}\backslash\{0\}$, so that (2.6) holds. If the manifold $\partial\mathcal{O}$ is analytic, then the map $\kappa_{int}$ will be analytic, and we can show that (2.6) will hold (see for instance [SV97]).

3. Tools for the proof of Proposition 4.1

Before proving Proposition 4.1, let us recall a few facts we will need in the proof.

3.1. An integral representation for the scattering amplitude

The operator

[TABLE]

can also be defined as follows. Let $v(\cdot;\xi,k)$ be the unique solutions to

[TABLE]

satisfying the Sommerfeld radiation condition.

$v$ may then be written as

[TABLE]

One can show (see for instance [HR76], page 381) that $A(k)$ is given by an integral kernel

[TABLE]

where $a$ satisfies

[TABLE]

3.2. The Kirchhoff approximation

The function $\partial_{\nu}v$ was studied in [MT85], where the authors write

[TABLE]

The main result we need from them can be summed up as follows. (The definition of the symbol classes $S_{\delta}$ is recalled in Appendix A.)

Theorem 3.1** (Melrose-Taylor, [MT85]).**

[TABLE]

where $\nu_{x}$ is the outward pointing normal vector at the point $x\in\partial\Omega$, and where $E$ satisfies

[TABLE]

In particular, we have

[TABLE]

We therefore have

[TABLE]

3.3. The use of Gaussian states

From now on, we fix a smooth compactly supported function

[TABLE]

taking value $1$ in a neighborhood of [math]. Let $(\omega_{0},\eta_{0})\in T^{*}\mathbb{S}^{d-1}$. We set

[TABLE]

Note that $\|\phi_{\omega_{0},\eta_{0}}\|_{L^{2}}=1+O(h^{\infty})$. The term \chi\big{(}k^{1/4}|\omega-\omega_{0}|\big{)} is not very important here, and we could replace it by \chi\big{(}k^{p}|\omega-\omega_{0}|\big{)} for any $p\in(0,1/2)$. It is here only to ensure that the integrals in (3.10) and (3.11) below makes sense.

The following lemma, whose proof can be found in [Ing16b, §4], says that any function can be decomposed along the $\phi_{\omega_{0},\eta_{0}}$ through a resolution of identity formula.

Lemma 3.2**.**

1) Let $f\in L^{2}(\mathbb{S}^{d-1})$. We have

[TABLE]

where $c_{k}$ is a parameter depending on $k$, with

[TABLE]

2) Let $T\in\mathcal{L}(L^{2}(\mathbb{S}^{d-1}))$ a trace class operator, then

[TABLE]

Let $\mathrm{d}$ be some Riemannian distance on $T^{*}\mathbb{S}^{d-1}$. For any $\varepsilon>0$, the set

[TABLE]

has volume $O(\varepsilon)$. Therefore, since $\|S(k)\|_{L^{2}\longrightarrow L^{2}}=1$, we have

[TABLE]

3.4. The action of the scattering matrix on a Gaussian state

The main tool we use in the proof of Theorem 1.1 is the following proposition, which describes the action of the scattering matrix on a Gaussian state

Proposition 3.3**.**

Let $(\omega_{0},\eta_{0})\in T^{*}\mathbb{S}^{d-1}\backslash G_{\varepsilon_{0}}$. Let us write $(\omega_{1},\eta_{1}):=\kappa(\omega_{0},\eta_{0})$. We have

[TABLE]

where

[TABLE]

$\Phi_{1}(\cdot;\omega_{0},\eta_{0})\in C^{\infty}(\mathbb{S}^{d-1};\mathbb{C})$* has a non-negative imaginary part vanishing only at $\omega_{1}$, and $\partial\Phi_{1}(\omega_{1};\omega_{0},\eta_{0})=\eta_{1}$, and $\partial^{2}\Phi_{1}(\omega_{1};\omega_{0},\eta_{0})$ has a positive definite imaginary part.*

Proof.

Thanks to (3.9, we have

[TABLE]

This is an oscillatory integral with a phase

[TABLE]

Let $y_{\pm}\in\partial\Omega$ be the points such that $y_{\pm}\in\eta_{0}+\mathbb{R}\omega_{0}$ and $\pm\nu_{y_{\pm}}\cdot\omega_{0}>0$. Let $\theta_{\pm}\in\mathbb{S}^{d-1}$ be such that $\omega_{0}-\theta_{\pm}\in\mathbb{R}\nu_{y_{\pm}}$ and $\theta_{\pm}\neq\omega_{0}$. We also set $\theta_{0}:=\omega_{0}$. Note that

[TABLE]

The phase $\Phi$ has a vanishing gradient and imaginary part at four points: $(\omega_{0},y_{+},\theta_{0})$, $(\omega_{0},y_{-},\theta_{0})$, $(\omega_{0},y_{+},\theta_{+})$ and $(\omega_{0},y_{-},\theta_{-})$.

We will show below that the second derivative of $\Phi$ is non-degenerate at these critical points. In particular, this implies thanks to Lemma A.1 that \big{(}(S_{k}-Id)\phi_{\omega_{0},\eta_{0}}\big{)}(\theta)=O(k^{-\infty}), unless, for some $j\in\{0,+,-\}$, we have $|\theta-\theta_{j}|=O_{\varepsilon}(k^{-1/2+\varepsilon})$.

We shall write

[TABLE]

so that $\delta_{\pm}(\theta_{\pm})=2\alpha_{\pm}$, and $\delta_{\pm}(\omega_{0})=0$.

Let us take a partition of unity on $\partial\Omega$:

[TABLE]

with $\chi_{\pm}\in C^{\infty}(\partial\Omega)$, satisfying $\chi_{\pm}(y)=1$ in a neighborhood of $y_{\pm}$. We shall consider the two integrals

[TABLE]

To analyse these integrals, let us introduce more convenient local coordinates.

Local Coordinates

We shall write the points $y\in\partial\Omega$ close to $y_{\pm}$ as

[TABLE]

where $u\in T_{y_{\pm}}\partial\Omega$, $f_{\pm}(u)\in\mathbb{R}\nu_{y_{\pm}}$, $f_{\pm}(u)=-\frac{u\cdot Mu}{2}\nu_{y_{0}}+O_{|u|\rightarrow 0}(|u|^{3})$. Here, $M$ is the second fundamental form of $\partial\Omega$ at $y_{\pm}$, and is therefore a positive definite symmetric matrix.

Similarly, we shall write the points $\omega\in\mathbb{S}^{d-1}$ close to $\omega_{0}$ as

[TABLE]

where $v\in\omega_{0}^{\perp}$, $g(v)\in\mathbb{R}\omega_{0}$, $g(v)=\frac{|v|^{2}}{2}\omega_{0}+O_{|v|\rightarrow 0}(|v|^{3})$.

Finally, for $j=0,+,-$, we shall write the points $\theta\in\mathbb{S}^{d-1}$ close to $\theta_{j}$, as

[TABLE]

where $w\in\theta_{j}^{\perp}$, $g(w)\in\mathbb{R}\theta_{j}$, $g(w)=\frac{|w|^{2}}{2}\theta_{j}+O_{|w|\rightarrow 0}(|w|^{3})$.

In these coordinates, using the fact that $(\theta_{j}-\omega_{0})\cdot u=0$, we have

[TABLE]

Since we are working away from the glancing set, the space $\omega_{0}^{\perp}\cap T_{y_{\pm}}\partial\Omega$ is of dimension $d-2$. By definition of $\theta_{j}$, we have

[TABLE]

for $j\in\{0,+,-\}$. Starting from an orthonormal basis of $\omega_{0}^{\perp}\cap T_{y_{\pm}}\partial\Omega$, we may find an orthonormal basis of $T_{y_{\pm}}\partial\Omega$, an orthonormal basis of $\omega_{0}^{\perp}$ and an orthonormal basis of $\theta_{j}^{\perp}$ such that, if $u$, $v$ and $w$ are written in the coordinates associated to these bases, we have

[TABLE]

where, notation as in (3.19),

[TABLE]

and where we denote by $\langle\cdot,\cdot\rangle_{n}$ the canonical scalar product on $\mathbb{R}^{n}$.

We therefore have

[TABLE]

where

[TABLE]

We may then write

[TABLE]

where $R\in S_{0}$ comes from the Jacobian of the change of coordinates $(u,v)\mapsto(y,\omega)$, and satisfies $R(0,0)=0$.

3.4.1. Behavior for $\theta$ close to $\omega_{0}$

The matrix $\mathcal{M}(\omega_{0})$ is invertible, with inverse

[TABLE]

and we have, notation as in (3.19),

[TABLE]

For $\theta$ close to $\omega_{0}$, $\mathcal{M}(\theta)$ is still invertible, with inverse

[TABLE]

By applying Lemma A.2, we obtain that

[TABLE]

Recall that we assume in this paragraph that |\theta-\theta_{0}|=O_{\varepsilon}\big{(}k^{-1/2+\varepsilon}\big{)} for all $\varepsilon>0$. Noting that $\theta\cdot y_{\pm}=\theta\cdot\eta_{0}+(\omega_{0}\cdot y_{\pm})(\theta\cdot\omega_{0})$, we deduce that

[TABLE]

Therefore, we have for all $\theta$ close to $\omega_{0}$.

[TABLE]

3.4.2. Behavior for $\theta$ close to $\theta_{\pm}$

Note that the matrix

[TABLE]

is invertible, since the second term is real symmetric, so that it does not have $(i-y_{\pm}\cdot\omega_{0})^{-1}$ in its spectrum. It is then straightforward to check that $\mathcal{M}(\theta)$ is invertible, with inverse

[TABLE]

Applying Lemma A.2, we obtain that for $|\theta-\theta_{\pm}|\leq k^{-1/2+\varepsilon}$, we have

[TABLE]

where

[TABLE]

Therefore, we have

[TABLE]

We therefore have

[TABLE]

where

[TABLE]

and $\Phi_{1}=\Phi_{1}^{-}$ satisfies the announced properties, thanks to (3.18). ∎

4. Proof of Theorem 1.1

The main ingredient in the proof of Theorem 1.1 is a trace formula for powers of $S_{k}$:

Proposition 4.1**.**

Suppose that (2.6) holds. Let $p\in\mathbb{Z}$. We have

[TABLE]

In particular, for any trigonometric polynomial $P$ vanishing at $1$ and for the measure $\mu_{k}$ in (1.3), as $k\to\infty$,

[TABLE]

Theorem 1.1 can be deduced from Proposition 4.1 in exactly the same way as in [Ing16a, §5]. We refer the reader to this paper for the argument.

Before proving Proposition 4.1, we shall prove the following corollary of Proposition 3.3.

Corollary 4.2**.**

Let $p\in\mathbb{N}$, let $(\omega_{0},\eta_{0})\in T^{*}\mathbb{S}^{d-1}\backslash G_{\varepsilon_{0}}$. Let us write $(\omega_{p},\eta_{p}):=\kappa^{p}(\omega_{0},\eta_{0})$. We have

[TABLE]

where for all $\varepsilon>0$,

[TABLE]

and $A_{1,p}(\theta;\omega_{0},\eta_{0},k)$ is such that we have

[TABLE]

for all $(\omega,\eta)\in T^{*}\mathbb{S}^{d-1}$ satisfying $\mathrm{d}((\omega,\eta),(\omega_{p},\eta_{p}))>k^{-1/2+\varepsilon}$ for some $\varepsilon>0$.

Proof.

We shall prove this result by induction. For $p=1$, the result is an immediate corollary of Proposition 3.3 and of the non-stationary phase Lemma A.1. Suppose that we have proven the result for some $p\in\mathbb{N}$, and let us prove it for $p+1$. By assumption, we have $S_{k}^{p+1}\phi_{\omega_{0},\eta_{0}}(\theta)=S_{k}A_{1,p}(\theta;\omega_{0},\eta_{0},k)+S_{k}A_{2,p}(\theta;\omega_{0},\eta_{0},k).$

By Lemma 3.2, we have

[TABLE]

where $c_{k}$ is equivalent to some power of $k$.

We therefore have

[TABLE]

We shall write

[TABLE]

In $A_{2,p+1}(\theta;\omega_{0},\eta_{0},k)$, the term $S_{k}A_{2,p}(\theta;\omega_{0},\eta_{0},k)$ is small in $L^{2}$ norm by recurrence hypothesis, since $S_{k}$ is unitary. For the other term, we note that the term $\langle A_{1,p}(\cdot;\omega_{0},\eta_{0},k),\phi_{\omega,\eta}\rangle$ in the integrand is $O(k^{-\infty})$ as soon as $(\omega,\eta)$ is at a distance larger than $k^{-1/2+\varepsilon}$ from $(\omega_{p},\eta_{p})$. Hence, the integrand is $O(k^{-\infty})$ away from a set of volume $O(k^{1-d+C\varepsilon})$ for some $C>0$. On this set, the integrand has an $L^{2}_{\theta}$ norm bounded by $O(k^{-1/3+\varepsilon})$. Therefore, $\|A_{2,p+1}\|_{L^{2}}=O(k^{-1/3+C^{\prime}\varepsilon})$.

As for $A_{1,p+1}$, we have

[TABLE]

Now, by the induction hypothesis, we know that the above integrand is $O(k^{-\infty})$ unless \mathrm{d}(\kappa(\omega^{\prime},\eta^{\prime}),(\omega_{p},\eta_{p}))=O\big{(}k^{-1/2+\varepsilon}\big{)} and \mathrm{d}((\omega,\eta),(\omega^{\prime},\eta^{\prime}))=O\big{(}k^{-1/2+\varepsilon}\big{)}. The result follows. ∎

We are now ready to prove Proposition 4.1.

Proof of Proposition 4.1.

First of all, let us note that it is enough to show the result for $p>0$. Indeed, since $S(k)$ is unitary, we have

[TABLE]

Therefore, let us fix from now on $p\geq 1$. By (3.11), we have

[TABLE]

By (3.13), the second term in the right-hand side of (4.2) is $O(\varepsilon_{0}k^{d-1}).$

To deal with the first term, we note that, when computing \big{(}S_{k}-Id\big{)}\phi_{\omega_{0},\eta_{0}}(\theta), the phase $\Phi$ satisfies $|\partial_{\omega}\Phi|=|\pi_{\omega}(\eta-y)|+|\omega-\omega_{0}|\geq C\mathrm{d}((\omega,\eta),\mathcal{I})$. Therefore, Lemma A.1 implies that we have

[TABLE]

Therefore, the first term in the right-hand side of (4.2) is $O(k^{-\infty}).$

We now compute

[TABLE]

Now, by Corollary 4.2, the integrand is $O(k^{-\infty})$ as soon as $\mathrm{d}(\kappa^{p}(\omega,\eta),(\omega,\eta))>k^{-1/2+\varepsilon}$. But, by (2.6), the volume of the set of $(\omega,\eta)$ satisfying $\mathrm{d}(\kappa^{p}(\omega,\eta),(\omega,\eta))>k^{-1/2+\varepsilon}$ is a $o_{k\rightarrow\infty}(1)$. All in all, we obtain that

[TABLE]

Therefore, (4.2) gives us

[TABLE]

and the result follows since this is true for all $\varepsilon_{0}>0$. ∎

5. Proof of theorem 1.3

We now give our alternative proof of the scattering phase asymptotics in Theorem 1.3. We begin by recalling that the scattering phase $s(k)$ can be defined continuously in such a way that $\lim_{k\to 0^{+}}s(k)=0$ and thus defined is in fact smooth for all $k>0$. We define the ‘reduced’ scattering phase by the sum

[TABLE]

where the logarithms of the eigenvalues, the $\beta_{k,n}$ are chosen to take values in $(-2\pi,0]$. For fixed $k$ the eigenvalues accumulate at $1$ from the bottom half plane and thus contribute positive values to the sum, which is nonetheless finite. A result of Eckmann-Pillet [EP95] shows that eigenvalues approach $1$ with positive imaginary part if and only if $k$ approaches a Dirichlet eigenvalue of $\Omega$. In fact, with $N_{D}(k)$ as in (1.7), we have

[TABLE]

Under the hypothesis that the measure of the periodic billiard trajectories in $\Omega$ is zero, it is known [Ivr80] that

[TABLE]

We claim that

[TABLE]

We will prove this by breaking up the unit circle into $M\in\mathbb{N}$ sectors of size $2\pi/M$ estimating the sum defining $s_{2\pi}(k)$ in these sectors. Namely, let $A_{M,k}(j):=\{-2\pi j/M<\beta_{k,n}\leq-2\pi(j+1)/M\}$ and $\alpha_{M,k}(j):=-\sum_{A_{M,k}(j)}\beta_{M,k}$, so that

[TABLE]

We begin with $j=0$, which is distinct from $j>0$ since there are infinitely many phase shifts in $A_{M,k}(0)$. We are going to show that

[TABLE]

Thanks to equation (2.3) in [Chr15] (which relies on the methods developed in [Zwo89]), we have that there exists $C>0$ independent of $k$ and $n$ such that

[TABLE]

Let us write

[TABLE]

Using (5.1) and a constant $C>0$ whose value changes from line to line, we see that

[TABLE]

For $j>0$, we estimate $\alpha_{M,k}(j)$ from above and below, and clearly

[TABLE]

It follows from the (1.2), since our sectors are size $2\pi/M$, that for $0<j\leq M-1$, for any $\delta>0$ and $k>k_{M,\delta}$, there is a constant $C>0$ such that

[TABLE]

Since $\sum_{j=1}^{M-1}(j+1)=M(M-1)/2$ we have

[TABLE]

and thus

[TABLE]

for any $M\in\mathbb{N},\delta>0$. Taking $\delta=1/M^{2}$ and sending $M\to\infty$ gives the result.

Appendix A Symbol classes and stationary phase

Let $X$ be a compact manifold, and let $a=(a_{k})_{k>0}$ be a family of $\mathbb{C}$-valued functions in $C^{\infty}(X)$, and let $0\leq\delta<1/2$. We shall write that $a\in S_{\delta}(X)$ if

[TABLE]

The following lemma follows from [Hör83, Lemma 7.7.1].

Lemma A.1**.**

Let $\delta<1/2$. Let $a\in S_{\delta}(X)$, and $\Phi\in S_{0}(X)$. Suppose that there exists $C,\varepsilon>0$ such that for all $x$ in the support of $a_{k}$, we have $|\partial\Phi_{k}(x)|\geq Ck^{-1/2+\varepsilon}$. We then have

[TABLE]

The following stationary phase result is a variant over [Hör83, Theorem 7.7.5], but we recall its proof for completeness. Note that we do not assume that the derivative of the phase vanishes at the origin, but only that $\partial\varphi_{k}(0)=O(k^{-1/2+\varepsilon})$ for all $\varepsilon>0$.

Lemma A.2**.**

Let $\varphi\in S_{0}(\mathbb{R}^{n})$ have a positive imaginary part, and be such that

• •

There exists $C>0$ such that for all $\varepsilon>0$, $\|\partial\varphi_{k}(x)\|=O(k^{-1/2+\varepsilon})\Leftrightarrow\|x\|=O(k^{-1/2+C\varepsilon})$.

• •

$\partial^{2}_{x}\varphi_{k}(0)$* is invertible for every $k$, and there exists $c>0$ independent of $k$ such that*

[TABLE]

Let $a\in S_{1/3}(\mathbb{R}^{n})$. Let us write $I(a):=\int_{\mathbb{R}^{n}}e^{ik\varphi_{k}(x)}a(x)\mathrm{d}x$. We have

[TABLE]

where the square-root of the determinant is defined as in [Hör83, §3.4].

In particular, if $a\in S_{0}(\mathbb{R}^{n})$, we have $a_{k}(\partial^{2}\varphi_{k}(0)^{-1}\partial\varphi_{k}(0))=a_{k}(0)+O(k^{-1/3})$, so that we have

[TABLE]

Proof.

Let $\chi\in C_{c}^{\infty}(\mathbb{R}^{n})$ be such that $\chi(x)=1$ if $|x|<1/2$, and $\varepsilon>0$. Set $\chi_{k}(x):=\chi(xk^{1/2-\varepsilon})$. We may write

[TABLE]

By Lemma A.1 and the assumption we made on $\varphi$, the second integral is $O(k^{-\infty})$.

We have

[TABLE]

and

[TABLE]

Indeed, all the derivatives of $\chi_{k}$ will vanish for such $x$, and, when differentiating e^{ik\big{(}\varphi_{k}(x)-\varphi_{k}(0)-x\cdot\partial\varphi_{k}(0)-\frac{1}{2}x\cdot\partial^{2}\varphi_{k}(0)x\big{)}} once, we get a factor of size $O(k^{\varepsilon})$. Differentiating again, each derivation makes the function grow at most of a factor $k^{1/2+\varepsilon}$.

Let us write b_{k}(x):=a_{k}(x)\chi_{k}(x)e^{ik\big{(}\varphi_{k}(x)-\varphi_{k}(0)-x\cdot\partial\varphi_{k}(0)-\frac{1}{2}x\cdot\partial^{2}\varphi_{k}(0)x\big{)}}, so that $b_{k}\in S_{1/2-\varepsilon}$. We have

[TABLE]

Let us write $A=A(k)=\partial^{2}\varphi(0)$, and $\xi_{0}=\xi_{0}(k)=\partial\varphi(0)$. We have

[TABLE]

so that, setting $y=x+A^{-1}\xi_{0}$, we have

[TABLE]

Writing $\tilde{b}_{k}(y)=b_{k}(y-A^{-1}\xi_{0})$ and $D=-i\partial$, we therefore have

[TABLE]

Now, we have

[TABLE]

so that by Plancherel’s equality,

[TABLE]

Thanks to [Hör83, Theorem 7.6.5], this quantity is equal to

[TABLE]

Since $b_{k}\in S_{1/2-\varepsilon}$ and $\|A\|^{-1}=O(1)$, the remainder can be made smaller than any power of $k^{-1}$ by taking $N$ large enough.

Using (A.2), we see that there exists $N\in\mathbb{N}$ such that

[TABLE]

The first term in the sum is $a(-A^{-1}\xi_{0})$, and the following ones are $O(k^{-1/3})$. The statement follows. ∎