Inverse problems in spectral geometry

TL;DR

This survey reviews recent advances in inverse spectral problems across various geometric and physical settings, highlighting how spectral data can determine geometric and operator properties.

Contribution

It compiles and discusses key results in inverse spectral and resonant problems for diverse operators and geometries, emphasizing recent progress and open questions.

Findings

Spectral data can uniquely determine certain geometric structures.

Inverse problems have been solved for specific classes of operators and domains.

The survey identifies gaps and future directions in spectral geometry research.

Abstract

In this survey we review positive inverse spectral and inverse resonant results for the following kinds of problems: Laplacians on bounded domains, Laplace-Beltrami operators on compact manifolds, Schr\"odinger operators, Laplacians on exterior domains, and Laplacians on manifolds which are hyperbolic near infinity.