Singular elliptic operators of second order with purely discrete spectra

Abstract

The Friedrichs extension of a second order singular elliptic operator is considered on a weighted L w 2 ( Ω ) \mathcal {L}_w^2(\Omega ) space. The region Ω \Omega is not necessarily bounded. Necessary conditions and sufficient conditions on the coefficients that will insure a discrete spectrum are given with a certain degree of sharpness achieved. The boundary conditions include the Dirichlet, Neumann, and mixed Dirichlet-Neumann boundary value problems.