Inverse problem for a general bounded domain in \(\mathbb{R}^3\) with piecewise smooth mixed boundary conditions
The author gives an asymptotic expansion formula for the trace of the heat kernel \(\Theta(t)= \sum^\infty_{\nu=1} \exp(-t\mu_\nu)\) as \(t\to 0^+\), where \(\mu_\nu\) denote the eigenvalues of the Helmholtz equation. The set of boundary conditions under consideration is fairly general including Dirichlet, Neumann, and Robin boundary conditions which can be simultaneously prescribed at different parts of the boundary. The coefficients of the expansion are expressed in terms of geometric quantities of the domain such as volume of the domain, surface area, mean and Gaussian curvature of the boundary. The result is applied to derive asymptotic expansions for certain thermodynamic quantities of an ideal gas for \(T\to\infty\), where \(T\) denotes the temperature of the gas. It is thus shown that geometric properties of the container have an influence on (quantum-)thermodynamic properties of the gas. The title and the abstract of the paper are slightly misleading, since the inverse aspect of the problem, i.e. determination of geometric properties from spectral information, is not addressed. Moreover, from the technical point of view, the paper is not entirely self-contained but relies in several points on results by the author [J. Math. Anal. Appl. 112, 455-470 (1985; Zbl 0617.35101)].
scientific article; zbMATH DE number 1529854