Introduction to spectral theory and inverse problem on asymptotically hyperbolic manifolds

Hiroshi Isozaki; Yaroslav Kurylev

Home/Archive/arXiv

arXiv · 1102.5382

Introduction to spectral theory and inverse problem on asymptotically hyperbolic manifolds

Hiroshi Isozaki·Yaroslav Kurylev

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.SP

Keep this discovery

BibTeX RIS

Hiroshi Isozaki, Yaroslav Kurylev. 2011-02-26. Introduction to spectral theory and inverse problem on asymptotically hyperbolic manifolds. https://arxiv.org/abs/1102.5382

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Regular hyperbolic tilings have no $\ell^2$ eigenfunctions

We show that the adjacency operator of the $1$-skeleton of any regular tiling of the hyperbolic plane has no nonzero square-integrable eigenfunctions. As a consequence, the same holds for every infinite connected regular graph admitting a proper planar embedding with regular dual.

math.SP↗

arXiv2026

Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension

This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(\Omega)$ and H\"older stability from terminal-time observations for sources in a suitable spectral Gevrey class.

math.SP↗

arXiv2026

Resolvent bounds and eigenvalue estimates of generalized Schr\"odinger operators with complex potentials on compact manifolds

We extend Cuenin's compact-manifold spectral bounds for Schr\"odinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators $P+V$, where $P$ is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and $V$ is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for $P$ imply $L^p$-$L^{p'}$ resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

math.SP↗