Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On quasi-isospectrality of potentials and Riemannian manifolds

Clara L. Aldana, Camilo Perez

TL;DR

This work generalizes isospectrality to the quasi-isospectral setting for Schrödinger and Laplace-type operators, blending expository synthesis with new constructions and rigidity results. It develops a Darboux-based double-commutation method to build quasi-isospectral Sturm–Liouville potentials on finite intervals and analyzes the impact of boundary conditions on the spectrum. The authors prove that quasi-isospectral closed manifolds of odd dimension are in fact isospectral and extend compactness results for isospectral potentials to the quasi-isospectral regime via heat-trace invariants, with complementary results in even dimensions. The paper also adapts heat-kernel asymptotics to quantify invariants across quasi-isospectral pairs and connects these findings to relative determinants, offering a cohesive framework bridging inverse spectral theory, Darboux transforms, and spectral geometry.

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.

On quasi-isospectrality of potentials and Riemannian manifolds

TL;DR

This work generalizes isospectrality to the quasi-isospectral setting for Schrödinger and Laplace-type operators, blending expository synthesis with new constructions and rigidity results. It develops a Darboux-based double-commutation method to build quasi-isospectral Sturm–Liouville potentials on finite intervals and analyzes the impact of boundary conditions on the spectrum. The authors prove that quasi-isospectral closed manifolds of odd dimension are in fact isospectral and extend compactness results for isospectral potentials to the quasi-isospectral regime via heat-trace invariants, with complementary results in even dimensions. The paper also adapts heat-kernel asymptotics to quantify invariants across quasi-isospectral pairs and connects these findings to relative determinants, offering a cohesive framework bridging inverse spectral theory, Darboux transforms, and spectral geometry.

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.
Paper Structure (20 sections, 181 equations)

This paper contains 20 sections, 181 equations.

Table of Contents

  1. Introduction
  2. Review of known rigidity results
  3. Sturm-Liouville operators on a finite interval
  4. Spectral rigidity of Schrödinger operators on compact manifolds.
  5. No rigidity of Sturm-Liouville with Dirichlet boundary conditions
  6. Quasi-isospectral potentials, after Bilotta, Morassi, and Turco
  7. Construction of the potential and the eigenfunctions
  8. The boundary conditions in Darboux's lemma
  9. The effect of Darboux's lemma on the spectrum
  10. Examples
  11. Example of a family of even quasi-isospectral potentials
  12. Examples with different boundary conditions
  13. Compactness of isospectral manifolds and potentials
  14. Compactness of isospectral Riemannian manifolds
  15. Compactness of isospectral potentials.
  16. ...and 5 more sections

Theorems & Definitions (5)

  • proof
  • proof
  • proof
  • proof
  • proof