Isospectrality and heat content

TL;DR

This paper demonstrates that isospectral operators, including various planar polygons and Schrödinger operators, can have different heat content, highlighting limitations of spectral data in determining heat-related properties.

Contribution

It provides explicit examples of isospectral operators with differing heat content, including polygons with mixed boundary conditions and Schrödinger operators, expanding understanding of spectral invariants.

Findings

Isospectral polygons can have different heat content.

Examples include polygons with infinitely many components.

Heat content is not determined solely by the spectrum.

Abstract

We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schr\"{o}dinger operators acting in $L^2[0,1]$ with Dirichlet boundary conditions, and show that an abundance of isospectral deformations do not preserve the heat content.