Introduction to Spectral Theory and Inverse Problem on Asymptotically Hyperbolic Manifolds

Abstract

Since −∆e ix·ξ = |ξ| 2 e ix·ξ , e ix·ξ is an eigenfunction of −∆. Therefore (0.1) and (0.2) illustrate the expansion of arbitrary functions in terms of eigenfunctions (more appropriately generalized eigenfunctions since they do not belong to L 2 (R n )) of the Laplacian.There are two directions of development of the above fact. One is quantum mechanics, where the Schrödinger operator H = −∆ + V (x) is the most basic tool to decribe the physical system of atoms or molecules. If H has the continuous spectrum, it is known that there exists a system of generalized eigenfunctions of H which play the same role as e ix·ξ . Moreover, by using these generalized eigenfunctions one can define an operator called the scattering matrix or the S-matrix, which is the fundamental object to study the physical properties of quantum mechanical particles through the scattering experiment.The other direction is the Fourier transform on manifolds, especially on homogeneous spaces of Lie groups, which is a central theme in the representation theory. Hyperbolic manifolds, one of the deepest sources of classical mathematics, appear also in this context. In particular, hyperbolic quotient manifolds by the action of discrete subgroups of SL(2, R) and the associated S-matrix are important objects in number theory. 0.2. Perturbation of the continuous spectrum. The aim of the perturbation theory of continuous spectrum is, given an operator H 0 whose spectral property is rather easy to understand, to study the spectral properties of H 0 + V , where V is the perturbation deforming the operator H 0 . When H = H 0 + V has the continuous spectrum, an effective way of studying its spectral properties is to construct a generalized Fourier tranform associated with H. To accomplish this idea, it is necessary that the Fourier transform for H 0 can be constructed easily. For example, it is the case for the Laplacian −∆ on R n . If the perturbation term V is an operator on the same Hilbert space as for H 0 and is not so strong, one can construct the Fourier transform associated with H 0 + V by using the technique of functional analysis and partial differential equations. This is not so easy for operators on hyperbolic manifolds. Even the construction of the Fourier transform associated with the Laplace-Beltrami operator on the hyperbolic space is no longer a trivial work. To construct the Fourier transform on hyperbolic spaces based on the upper half space model or the ball model, one needs deep knowledge of Bessel functions. Under the action of discrete subgroups, the properties of groups will reflect on the structure of manifolds or the construction of generalized eigenfunctions. 0.3. Spectral and scattering theory on hyperbolic manifolds. In the present note, we deal with the spectral theory and the associated forward and inverse problems for Laplace-Beltrami operators on hyperbolic manifolds. Since we are mainly interested in its spectral properties, Selberg's work [Se56] and its developments are beyond our scope. As an approach to the hyperbolic man...

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

“…In [42], it is shown that the asymptotics of the metric of an AHM are uniquely determined (up to isometries) by the scattering matrix S g (λ) at a fixed energy λ off a discrete subset of R. In [65], it is proved that the metric of an AHM is uniquely determined (up to isometries) by the scattering matrix S g (λ) for every λ ∈ R off an exceptional subset. Similar results are obtained recently in [40] for even more general classes of AHM. In [35], it is proved that, for connected conformally compact Einstein manifolds of even dimension n + 1, the scattering matrix at energy n on an open subset of its conformal boundary determines the manifold up to isometries.…”

“…We recall that inverse scattering problems at fixed energy on asymptotically hyperbolic manifolds are closely related to the anisotropic Calderón problem on compact riemannian manifolds with boundary. We refer to the surveys [38,40,46,48,66,71] for the current state of the art on this question. One of the aim of this paper is thus to give examples of manifolds on which we can solve the inverse scattering problem at fixed energy but do not have one of the particular structures for which the uniqueness for the anisotropic Calderón problem on compact manifolds with boundary is known.…”

“…We recall here the construction of the scattering operator given in [40,41] for asymptotically hyperbolic manifolds. This construction has been used in [22] in the case of asymptotically hyperbolic Liouville surfaces.…”

“…For general Asymptotically Hyperbolic Manifolds (AHM in short) with no particular (hidden) symmetry, direct and inverse scattering results for scalar waves have been proved by Joshi and Sá Barreto in [42], by Sá Barreto in [65], by Guillarmou and Sá Barreto in [35,36] and by Isozaki and Kurylev in [40]. In [42], it is shown that the asymptotics of the metric of an AHM are uniquely determined (up to isometries) by the scattering matrix S g (λ) at a fixed energy λ off a discrete subset of R. In [65], it is proved that the metric of an AHM is uniquely determined (up to isometries) by the scattering matrix S g (λ) for every λ ∈ R off an exceptional subset.…”

Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces

“…The Liouville surfaces S we are considering in this paper thus possess two asymptotically hyperbolic ends at {x = 0} and {x = A} as well as this remarkable property of separability for the wave equation mentioned above. We refer to [3,11,12,13] for a detailed presentation of general asymptotically hyperbolic surfaces (and higher dimensional asymptotically hyperbolic manifolds) with no hypothesis of (hidden) symmetries or integrability of the geodesic flow. We start defining the main object from which we shall study some inverse problems at fixed energy for such asymptotically hyperbolic Liouville surfaces.…”