Let \((M,g)\) be a compact \(m\)-dimensional Riemannian manifold with boundary \(\partial M\). Let \(\text{Spec}(M,g)\) be the collection of the eigenvalues of the Laplace-Beltrami operator with Dirichlet boundary conditions where each eigenvalue is repeated according to multiplicity. Two manifolds \((M_i,g_i)\) are said to be isospectral if \(\text{Spec}(M_1,g_1)=\text{Spec}(M_2,g_2)\). Let \(u(x;t)\) be the temperature of \((M,g)\) with Dirichlet boundary conditions and initial temperature \(1\). This is the solution of the equation \(\Delta u+\partial_tu=0\), \(u|_{\partial M}=0\), and \(\lim_{t\downarrow0}u(\cdot,t)=1\) in \(L^2\). Let \(E(t)=\int_Mu(x;t)dx\) where \(dx\) is the Riemannian measure on \(M\). This is the total heat energy content.
The authors present examples of planar polygons which are isospectral but which do not have the same total heat energy content; these include examples with infinitely many components. Other examples are presented where mixed Dirichlet and Neumann boundary conditions are used. The authors also consider Schrödinger operators on \([0,1]\) and present isospectral deformations which do not preserve the heat content.
§1 of the paper presents an introduction to the matter at hand. In §2 a variety of examples where explicit computations are possible are discussed and it is shown that isospectral does not imply isoheat for open sets in Euclidean space. Both the short time heat content asymptotics (which are locally computable) and the large-time behavior (which is not locally computable) is used. In §3, the operators discuss Schrödinger operators \(-\partial_x^2+q\) on \([0,1]\) with Dirichlet boundary conditions at \(0\) and \(1\) where \(q\in L^2[0,1]\).