An Inverse Spectral Theorem

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.

15 pages. Keywords: inverse problems, Ambarzumyan, spectral geometry, heat kernel, heat trace asymptotics, quantum graph