The spectral shift function for planar obstacle scattering at low energy

Let $H$ signify the free non-negative Laplacian on $mathbb{R}^2$ and $H_Y$ the non-negative Dirichlet Laplacian on the complement $Y$ of a nonpolar compact subset $K$ in the plane. We derive the low-energy expansion for the Krein spectral shift function (scattering phase) for the obstacle scattering system ${,H_Y,,H,}$ including detailed expressions for the first three coefficients. We use this to investigate the large time behaviour of the expected volume of the pinned Wiener sausage associated to $K$.