Equidistribution of phase shifts in trapped scattering

Abstract

We consider semiclassical scattering for compactly supported perturbations of the Laplacian and show equidistribution of eigenvalues of the scattering matrix at (classically) non-degenerate energy levels. The only requirement is that sets of fixed points of certain natural scattering relations have measure zero. This extends the result of [GRHZ15], where the authors proved the equidistribution result under a similar assumption on fixed points but with the condition that there is no trapping. IntroductionConsider a Riemannian manifold (X, g) which is Euclidean near infinity, in the sense that there exist compact sets X 0 ⊂ X and K 0 ⊂ R d such that (X\X 0 , g) and (R d \K 0 , g eucl ) are isometric.Let us consider an operatorWe define the scattering matrix 1 S h : C ∞ (S d−1 ) −→ C ∞ (S d−1 ), which depends on h, by S h (φ in ) := e iπ(d−1)/2 φ out .The factor e iπ(d−1)/2 is taken so that the scattering matrix is the identity operator when (X, g) = (R d , g Eucl ) and V ≡ 0.For each h ∈ (0, 1], S h can be extended by density to a unitary operator acting on L 2 (S d−1 ). S h − Id is then a trace class operator. Therefore, S h admits a sequence of eigenvalues of modulus 1, which converge to 1, and which we denote by (e iβ h,n ) n∈N .Our aim in this paper will be to study the behaviour of (e iβ h,n ) in the limit where h → 0. To do this, we define a measure µ h on S 1 byfor any continuous f : S 1 −→ C. This measure is not finite, but µ h , f is finite as soon as 1 is not in the support of f .Let us now state the assumptions we make on the manifold X and on the potential V .

Equidistribution of phase shifts in obstacle scattering

“…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.…”

“…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].…”

The semiclassical structure of the scattering matrix for a manifold with infinite cylindrical end

“…A point y ∈ B is in the domain D κ if and only if the trajectory t → Φ t (c, −H(y); y) is not forward-trapping. By(29),(33) Φ t (c, −H(y); y) = (c − 2tH(y), −H(y) ; Ψ −2tH(y) (y)).Therefore this trajectory traverses the hypersurface {r = 0} at time t = c 2H(y) and at the point (0, −H(y); Ψ −c (y)). It follows that if we letϑ := Ψ −c : B → B, then ϑ maps D κ into D κ .…”

“…The distribution of phase shifts has been studied in a number of Euclidean settings, e.g. [6,16,20,29,21,19]. Some of these papers use the results of Alexandrova or Ingremeau on the microlocal structure of the scattering matrix.…”