Scattering Phases and Density of States for Exterior Domain

For a bounded open domain $\mathrm{\Omega }\in {\mathrm{\Re }}^2$ with connected complement and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\text{DO}$ on $\mathrm{\Omega }$ and the S-matrix on the complement ${\mathrm{\Omega }}^c$. Using the restriction $A_E$ of $(-\mathrm{\Delta }-E{)}^{-1}$ to the boundary of $\mathrm{\Omega }$, we establish that $A_{E_0}^{-1\mathrm{/}2}{A}_E{A}_{E_0}^{-1\mathrm{/}2}-1$ 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.