The spectral shift function for planar obstacle scattering at low energy

TL;DR

This paper derives the low-energy expansion of the spectral shift function for planar obstacle scattering and explores its implications for the long-time behavior of the Wiener sausage volume.

Contribution

It provides explicit formulas for the first three coefficients in the low-energy expansion of the spectral shift function for obstacle scattering in the plane.

Findings

Explicit low-energy expansion coefficients derived.

Analysis of Wiener sausage volume growth at large times.

Connections between spectral shift and probabilistic geometric quantities.

Abstract

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 $K$.