The Steklov spectrum of the Helmholtz operator

The Steklov eigenvalue problem is a boundary value problem where the spectral parameter relates the Dirichlet and Neumann traces. It has been extensively studied for the Laplacian. In this thesis, we study the Steklov problem for the Helmholtz operator −∆ − µ2 with real wave number µ for bounded planar domains. After reformulating this problem as a boundary integral equation we implement a numerical method with the help of layer potentials approximated using appropriate quadratures. We obtain a generalized eigenvalue problem for the Steklov-Helmholtz eigenvalues and eigendensities, where the eigenfunctions are reconstructed using the single layer potential. We describe a numerical approach for the reformulated problem. We observe exponential convergence of the numerically computed Steklov-Helmholtz eigenvalues. A Weyl-type law is observed for the asymptotics of the spectrum. We report on additional experiments including a shape optimization problem. The numerical scheme is tested on a variety of domains of genus 0 and 1. Our approach can be adapted to other problems as well.