Scattering phases and density of states for exterior domains

For a bounded open domain $\Omega\subset\mathbb R^2$ with connected complement and piecewise smooth boundary, we consider the Dirichlet Laplacian $\Delta$ on $\Omega$ and the S-matrix on its complement. We obtain precise bounds on the total scattering phase and and a Krein spectral formula, which improve similar results found in the literature