Low energy scattering asymptotics for planar obstacles

TL;DR

This paper derives low energy asymptotics for the scattering problem involving planar obstacles, revealing how the obstacle's properties influence scattering behavior at low energies.

Contribution

It introduces a method to compute low energy asymptotics for the resolvent and scattering matrix for planar obstacles, extending understanding of scattering in two dimensions.

Findings

Asymptotics expressed in terms of obstacle's logarithmic capacity

Relation between obstacle resolvent and free resolvent established

Results expected to generalize to broader perturbations of the Laplacian

Abstract

We compute low energy asymptotics for the resolvent of a planar obstacle, and deduce asymptotics for the corresponding scattering matrix, scattering phase, and exterior Dirichlet-to-Neumann operator. We use an identity of Vodev to relate the obstacle resolvent to the free resolvent and an identity of Petkov and Zworski to relate the scattering matrix to the resolvent. The leading singularities are given in terms of the obstacle's logarithmic capacity or Robin constant. We expect these results to hold for more general compactly supported perturbations of the Laplacian on $\mathbb R^2$, with the definition of the Robin constant suitably modified, under a generic assumption that the spectrum is regular at zero.