Scattering Phases and Density of States for Exterior Domain

Abstract

For a bounded open domain $\Omega \in {\Re }^2$ with connected complement and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\DO$ on $\Omega$ and the S-matrix on the complement ${\Omega }^c$. Using the restriction $A_E$ of $(-\Delta -E{)}^{-1}$ to the boundary of $\Omega$, we establish that $A_{E_0}^{-1/2}{A}_E{A}_{E_0}^{-1/2}-1$ is trace class when $E_0$ is negative and give bounds on the energy dependence of this difference. This allows for precise bounds on the total scattering phase, the definition of a $\zeta$-function, and a Krein spectral formula, which improve similar results found in the literature.

arXiv comment15 pages, Postscript, A4