Martin Nilsson: Faber-Krahn inequalities for point-interaction Hamiltonians in bounded domains

Abstract

This thesis concerns the minimization of the lowest eigenvalue of certain perturbations of the Dirichlet Laplacian on a bounded domain with potential supported on a discrete set of points. Recent results include a Faber-Krahn-type inequality, valid when the support is a single point. We show that some of the methods used to prove this inequality can be generalized to an abstract operator-theoretic setting, and we use this to obtain a new Faber-Krahn-type inequality for when the support consists of more than one point. Our approach also yields some additional results on the lowest eigenvalue, such as it being simple.

Zoom notes: Password required, contact arias@math.su.se