Some spectral properties of Rooms and Passages domains and their skeletons

Abstract

In this paper we investigate spectral properties of Lapla- cians on Rooms and Passages domains. In the first part, we use Dirichlet- Neumann bracketing techniques to show that for the Neumann Lapla- cian in certain Rooms and Passages domains the second term of the asymptotic expansion of the counting function is of order $\sqrt{\lambda }$. For the Dirichlet Laplacian our methods only give an upper estimate of the form $\sqrt{\lambda }$. In the second part of the paper, we consider the relation- ship between Neumann Laplacians on Rooms and Passages domains and Sturm-Liouville operators on the skeleton.

19 pages, 6 figures, to appear in Proceedings of Symposia in Pure Mathematics