Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds

Abstract

In this paper, we adapt the well-known local uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schrödinger equation to prove local uniqueness results in the setting of inverse metric problems. More specifically, we consider a class of spherically symmetric manifolds having two asymptotically hyperbolic ends and study the scattering properties of massless Dirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationary scattering is encoded by a countable family of one-dimensional Dirac equations. This allows us to define the corresponding transmission coefficients T (λ, n) and reflection coefficients L(λ, n) and R(λ, n) of a Dirac wave having a fixed energy λ and angular momentum n. For instance, the reflection coefficients L(λ, n) correspond to the scattering experiment in which a wave is sent from the left end in the remote past and measured in the same left end in the future. The main result of this paper is an inverse uniqueness result local in nature. Namely, we prove that for a fixed λ = 0, the knowledge of the reflection coefficients L(λ, n) (resp. R(λ, n)) -up to a precise error term of the form O(e −2nB ) with B > 0 -determines the manifold in a neighbourhood of the left (resp. right) end, the size of this neighbourhood depending on the magnitude B of the error term. The crucial ingredients in the proof of this result are the Complex Angular Momentum method as well as some useful uniqueness results for Laplace transforms.

Quasi-normal modes for Dirac fields in the Kerr–Newman–de Sitter black holes

“…Moreover, we get similar results for the wave equation in de Sitter-Schwarzschild metric thus improving the results in [4]. Our results extend to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds (see [13]).…”

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

“…This same approach has been used recently to study scattering inverse problems for asymptotically hyperbolic manifolds (see [11,12,10]). In the hyperbolic setting, a Liouville transformation changes the angular momentum variable in a spectral variable, and we can see the Jost solutions as suitable perturbations of the modified Bessel functions I ν (z).…”

“…Actually, we emphasize that we have a better result; it suffices to have the previous estimate for ν integer, (see [11], Proposition 4.2). We prefer to give here this result in this discrete setting since we shall use it again in the next Section.…”

“…Moreover, this solution, called the regular solution, has the asymptotic expansion at infinity: 11) where the constant c(l) is the modulus of the Jost function β(ν(l)), (see Section 2 for details).…”

Inverse Scattering at Fixed Energy on Three-Dimensional Asymptotically Hyperbolic Stäckel Manifolds

“…We also refer to [1,61] for books dealing with this method. This tool was already used in the field of inverse problems for one angular momentum in [21,22,23,24,25,26,34,62] and we note that this method is also used in high energy physics (see [20]). In this work we use a novel multivariable version of the Complexification of the Angular Momentum method for two angular momenta which correspond to the constants of separation of the Helmholtz equation.…”