On quasi-isospectrality of potentials and Riemannian manifolds

TL;DR

This paper explores the concept of quasi-isospectral operators, extending classical isospectral theory, and demonstrates that quasi-isospectral closed manifolds of odd dimension are actually isospectral, with implications for spectral geometry.

Contribution

It introduces the notion of quasi-isospectrality, applies the BMT method to construct such operators, and extends classical results to this new setting.

Findings

Quasi-isospectral manifolds of odd dimension are necessarily isospectral.

The BMT method effectively constructs quasi-isospectral Sturm-Liouville operators.

Classical compactness results extend to the quasi-isospectral context via heat trace asymptotics.

Abstract

In this article, we study quasi-isospectral operators as a generalization of isospectral operators. The paper contains both expository material and original results. We begin by reviewing known results on isospectral potentials on compact manifolds and finite intervals, and then introduce the notion of quasi-isospectrality. We next investigate the BMT method as a systematic approach to constructing quasi-isospectral Sturm-Liouville operators on a finite interval, and apply it to several boundary value problems. Our main result shows that any two quasi-isospectral closed manifolds of odd dimension are, in fact, isospectral. In addition, we extend classical compactness results for isospectral potentials on low-dimensional manifolds to the quasi-isospectral setting via heat trace asymptotics.