[Paper Review] Inverse problems in spectral geometry

Kiril Datchev, Hamid Hezari|arXiv (Cornell University)|Aug 29, 2011

Spectral Theory in Mathematical PhysicsMathematics86 references20 citations

TL;DR

This paper surveys inverse spectral problems in geometry, focusing on whether the spectrum of the Laplace-Beltrami operator determines the geometry of a domain or manifold up to isometry. It synthesizes results on heat and wave trace invariants, showing that under analyticity and symmetry assumptions, spectral data can uniquely determine shapes like ellipses, spheres, and manifolds with integrable or chaotic geodesic flow, while also highlighting open problems and counterexamples.

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, Schrödinger operators, Laplacians on exterior domains, and Laplacians on manifolds which are hyperbolic near infinity.

Motivation & Objective

• To understand which geometric invariants—such as area, perimeter, Euler characteristic, or topology—are determined by the spectrum of the Laplacian on domains and manifolds.
• To investigate whether geometric objects like ellipses, spheres, or flat manifolds are spectrally unique under symmetry and analyticity assumptions.
• To examine the role of wave trace invariants and heat invariants in reconstructing geometry from spectral data.
• To identify conditions under which isospectral deformations are trivial (spectral rigidity) or compact (spectral compactness).
• To clarify the limitations of spectral invariants by constructing examples with identical invariants but non-isospectral manifolds.

Proposed method

• Uses heat trace asymptotics $\operatorname{Tr}e^{-t\Delta} \sim \sum a_j t^{j/2}$ as $t \to 0^+$, where coefficients $a_j$ are geometric invariants (e.g., $a_0 = \text{Area}/(2\pi)$, $a_1 = \text{Perimeter}/(4\pi)$).
• Applies wave trace expansions $\operatorname{Tr}\cos(t\sqrt{\Delta})$ near $t = T$, where $T$ is the length of a periodic geodesic, to extract local invariants from periodic orbits.
• Employs quantum Birkhoff normal forms to express wave invariants $b_j$ as integrals of curvature and metric derivatives along periodic geodesics.
• Leverages nondegenerate periodic orbits and their iterations to derive uniqueness results via wave trace invariants in analytic or symmetric settings.
• Uses Gutzwiller's trace formula for Schrödinger operators in the semiclassical limit to analyze spectral invariants in quantum systems.
• Constructs counterexamples using perturbations of symmetric potentials (e.g., harmonic oscillator) to show agreement of invariants up to $\mathcal{O}(h^\infty)$ despite differing spectra.

Experimental results

Research questions

• RQ1 Can the area, perimeter, and Euler characteristic of a domain be recovered from its Laplace eigenvalue spectrum?
• RQ2 Are ellipses and spheres spectrally rigid, i.e., uniquely determined by their spectrum among isospectral deformations?
• RQ3 To what extent do wave trace invariants—derived from periodic geodesics—determine the geometry of a manifold?
• RQ4 Can two non-isometric manifolds have identical heat or wave trace invariants, and if so, what are the conditions?
• RQ5 What is the role of analyticity and symmetry in ensuring spectral uniqueness, and how do these assumptions affect the strength of inverse results?

Key findings

• The area and perimeter of a domain are spectral invariants, as shown by Weyl's asymptotic law and Pleijel's refinement of the heat trace.
• The Euler characteristic is a spectral invariant, with $a_2$ in the heat trace expansion giving $\chi(\Omega)$, as proved by McKean and Singer.
• There exist isospectral but non-isometric domains (e.g., Gordon-Webb-Wolpert domains), but such examples are non-convex and non-smooth, leaving the spectral rigidity of convex or smooth domains open.
• For real analytic and symmetric domains, spectral uniqueness holds under mild nondegeneracy conditions, relying on wave trace invariants from periodic orbits.
• Spectral rigidity is established for ellipses, spheres, flat manifolds, and constant negative curvature manifolds due to special features of their geodesic flow.
• It remains an open problem to construct two non-isospectral manifolds with identical wave trace invariants, though such examples are conjectured to exist.