Mathematics – Spectral Theory
Scientific paper
2011-02-26
Mathematics
Spectral Theory
Scientific paper
We study spectral properties of the Laplace-Beltrami operator on asymptotically hyperbolic manifolds and their applications to inverse scattering. We deal with the general short-range perturbation of the metric. The main part of the monograph deals with the direct problem, namely, (1) Location of the essential spectrum. (2) Absence of eigenvalues embedded in the continuous spectrum. (3) Discreteness of embedded eigenvalues in the continuous spectrum when all the ends are cusps. (4) Limiting absorption principle for the resolvent and the absolute continuity of the continuous spectrum. (5) Construction of the generalized Fourier transform. (6) Asymptotic completeness of time-dependent wave operators. (7) Characterization of the space of scattering solutions to the Helmhotz equation in terms of the generalized Fourier transform. (8) Asymptotic expansion of scattering solutions to the Helmholtz equation and the S-matrix. As a byproduct, we also study (9) Representation of the fundamental solution to the wave equation in the upper-half space model. (10) Radon transform and the propagation of singularities for the wave equation. Finally, we discuss the inverse scattering problem of (11) Identification of the Riemannian metric from the scattering matrix.
Isozaki Hiroshi
Kurylev Yaroslav
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