Isospectrality and Heat Content
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 Schrodinger 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.
Bulletin of the London Mathematical Society
van den Berg, M.; Dryden, Emily; and Kappeler, T.. "Isospectrality and Heat Content." Bulletin of the London Mathematical Society (2014) : 793-808.