An Inverse Spectral Theorem

TL;DR

This paper extends an inverse spectral theorem to a broader class of geometric and graph structures, providing new tools for spectral analysis on manifolds and graphs with Neumann or Kirchhoff boundary conditions.

Contribution

It generalizes Ambarzumyan's inverse spectral theorem to arbitrary compact Riemannian manifolds, quantum graphs, and combinatorial graphs with specific boundary conditions.

Findings

Extended inverse spectral theorem applicable to various structures

Applicable to Riemannian manifolds, quantum graphs, and combinatorial graphs

Provides new methods for spectral analysis under Neumann or Kirchhoff conditions

Abstract

We prove a substantial extension of an inverse spectral theorem of Ambarzumyan, and show that it can be applied to arbitrary compact Riemannian manifolds, compact quantum graphs and finite combinatorial graphs, subject to the imposition of Neumann (or Kirchhoff) boundary conditions.