Hearing the shape of membranes: Further results (Q2638447)

The spectral function \(\theta (t)=\sum \exp (-t\lambda_ m)\), \(t>0\), where \(\{\lambda_ m\}\) are the eigenvalues of the Laplacian in \({\mathbb{R}}^ n\), \(n=2\) or 3, is studied for a variety of domains, especially relative to the boundary condition (see impedance tomography) \(\partial u/\partial n+\gamma_ ju=0\) on \(\Gamma_ j\) or \(S_ j\), \(j=1,...,J\). Here \(\Gamma_ j\subset {\mathbb{R}}^ 2\) and \(S_ j\subset {\mathbb{R}}^ 3\) are parts of the surface of a ball. \(\theta\) is represented by means of Green's function of the heat equation relative to these boundary conditions. There are generalizations to other domains.