Isospectrality and heat content

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 Schrödinger 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.Mathematics Subject Classification (2010): 58J53; 58J50; 35P10.

Heat content for convolution semigroups

“…In paper [19], the author calls the quantity H Ω (t) heat content and we will use the same terminology. There are a lot of articles where bounds and asymptotic behaviour of the heat content related to Brownian motion, either on R d or on compact manifolds, were studied, see [19], [21], [22], [20], [18], [23]. Recently Acuña Valverde [2] investigated the heat content for isotropic stable processes in R d , see also [1] and [3].…”

Transient response functions for graph structure addressable memory

“…For large and complex graphs the intrinsic parallelism as well as the transient behavior comparison enable fast similarity testing. Since the motion modes of the random walk and their strength are governed by the spectral resolution (the eigenvalues and the eigenvectors) of the graphs [26], it is meaningful to use a similarity criterion based on the spectral resolution. A simple example is the similarity testing of graphs that represent independently generated images of white noise.…”

“…The partial differential equation above can be subjected to the spectral decomposition that leads to the method of obtaining the spectral resolution information of the domain, as mentioned before. In fact people have studied the heat content function, defined as g(t) = D u(x, t)dx, for obtaining the relevant information about the domain in, e.g., [27], [22], [26]. When V (x) = 0 one uses Duhamel's principle for the superposition over time and the above thought carries over.…”

“…Can quickness be achieved via zigzag random walk paths? The "small-time asymptotics" results in [27], [22], [26] suggest that this is possible. Historically the small-time asymtpotics about diffusion was considered by the famous mathematician Mark Kac.…”

“…However it is worth noting that [10], [26] give examples where "isoheat" (namely h(t) are the same) does not imply isospetral (the eigenvalues are the same). These examples involve unusual constructions and should not concern us for our purposes.…”