On the eigenvalues of the Robin Laplacian with a complex parameter

Abstract

We are interested in the eigenvalues of the Laplacian on a bounded domain with boundary conditions of the form $\frac{\partial u}{\partial \nu }+\alpha u=0$, where $\nu$ is the outer unit normal to the boundary and $\alpha$ should be considered a parameter on which the eigenvalues depend.

For positive $\alpha$ this operator, and in particular its eigenvalues, interpolate in a strong sense between those of the Neumann ($\alpha =0$) and Dirichlet (formally $\alpha =\infty$) Laplacians. In recent years, however, the case of large negative $\alpha$ has been studied intensively, and in particular the asymptotics of the eigenvalues in the singular limit $\alpha \to -\infty$ is well understood: there is a sequence of eigenvalues which diverges like $-{\alpha }^2$, independently of the geometry of the domain, while any non-divergent eigenvalues converge to points in the spectrum of the Dirichlet Laplacian.

Here, after giving a brief overview of what is known for real $\alpha$, we will present a number of new results for the corresponding problem when $\alpha$ is a (usually large) complex parameter. This is based on ongoing joing work with Sabine Bögli (Imperial College London) and Robin Lang (University of Stuttgart).