Scattering Phases and Density of States for Exterior Domain

Abstract

For a bounded open domain $Ω\in \real^2$ with connected complement and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\DO$ on $Ω$ and the S-matrix on the complement $Ω^c$. Using the restriction $A_E$ of $(-Δ-E)^{-1}$ to the boundary of $Ω$, we establish that $A_{E_0}^{-1/2}A_EA_{E_0}^{-1/2}-1$ 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 $ζ$-function, and a Krein spectral formula, which improve similar results found in the literature.