  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 $K$.
