Scattering for the Dirichlet-to-Neumann map and the Steklov spectrum

Abstract

We develop scattering theory for Dirichlet-to-Neumann (DtN) maps on asymptotically conic, unbounded domains $Ω\subset \mathbb{R}^d$, that are only Lipschitz inside a compact set and whose boundary extends to infinity. Specifically, we prove a limiting absorption principle and demonstrate existence of the scattering matrix. As an application of the theory, we show that the eigenvalues of Steklov-type problems on curvilinear polygons, $\mathcal{P}$, are well approximated by the eigenvalues of a Dirac operator on $\partial\mathcal{P}$ whose transmission coefficients between edges meeting at a corner are given by the scattering matrix for DtN maps on asymptotically conic domains associated to the corners. Up to an arbitrary loss in the power of the eigenvalue, the error in the approximation is optimal among all possible asymptotics that use only local information about $\partial\mathcal{P}$.