Non-Friedrichs Self-adjoint extensions of the Laplacian in R^d

Abstract

This is an expository paper about self-adjoint extensions of the Laplacian on R^d, initially defined on functions supported away from a point. Let L be the Laplacian with domain smooth functions with compact support away from the origin. We determine all the self-adjoint extensions of L by calculating the deficiency subspaces of the closure of L. In the case d=3, the self-adjoint extensions are parametrized by a circle, or equivalently by the real line plus a point at infinity. We show that the non-Friedrichs extensions have either a single eigenvalue or a single resonance. This is of some interest since self-adjoint Schr\"odinger operators $\Delta+V$ on R^d either have no resonance (if $V\equiv 0$) or infinitely many (if $V\not\equiv 0$). For these non-Friedrichs self-adjoint extensions, the kernel of the resolvent is calculated explicitly. Finally we use this knowledge to study a wave equation involving these non-Friedrichs extensions, and explicitly determine the wave kernel.