Hearing the shape of a drum by knocking around

TL;DR

This paper investigates whether the shape of a drum can be uniquely determined by local spectral information at every point, showing that it is possible if the spectrum is simple, and providing counterexamples otherwise.

Contribution

It demonstrates that the local Weyl counting function at each point suffices to recover the Riemannian metric when the spectrum is simple, extending Kac's classic question.

Findings

Unique recovery of the shape when the spectrum is simple

Counterexamples showing necessity of the simplicity condition

Extension of Kac's question to local spectral data

Abstract

We study a variation of Kac's question, "Can one hear the shape of a drum?" if we allow ourselves access to some additional information. In particular, we allow ourselves to ``hear" the local Weyl counting function at each point on the manifold and ask if this is enough to uniquely recover the Riemannian metric. This is physically equivalent to asking whether one can determine the shape of a drum if one is allowed to knock at any place on the drum. We show that the answer to this question is ``yes" provided the Laplace-Beltrami spectrum of the drum is simple. We also provide a counterexample illustrating why this hypothesis is necessary.