Introduction to spectral theory and inverse problem on asymptotically hyperbolic manifolds

TL;DR

This paper explores the spectral properties and inverse scattering problems on asymptotically hyperbolic manifolds, including spectrum location, wave operators, and metric reconstruction from scattering data.

Contribution

It provides new insights into spectral analysis, wave operator completeness, and inverse problems for asymptotically hyperbolic manifolds, advancing understanding of their geometric and spectral structure.

Findings

Location of the essential spectrum established

Absence of embedded eigenvalues proven

Construction of the generalized Fourier transform achieved

Abstract

We study the spectral theory and inverse problem on asymptotically hyperbolic manifolds. The main subjects are as follows: (1)Location of the essential spectrum. (2)Absence of eigenvalues embedded in the continuous spectrum. (3)Limiting absorption principle for the resolvent and the absolute continuity of the continuous spectrum. (4)Construction of the generalized Fourier transform. (5)symptotic completeness of time-dependent wave operators. (6)Characterization of the space of scattering solutions to the Helmhotz equation in terms of the generalized Fourier transform. (7)Asymptotic expansion of scattering solutions to the Helmholtz equation and the S-matrix. (8)Representation of the fundamental solution to the wave equation in the upper-half space model. (9)Radon transform and the propagation of singularities for the wave equation. Finally, we shall discuss the inverse problem. Namely (10)Identification of the Riemannian metric from the scattering matrix.