Schrödinger operators with Leray-Hardy potential singular on the boundary

Abstract

We study the kernel function of the operator u↦Lμu=−Δu+μ|x|2u in a bounded smooth domain Ω⊂R+N such that 0∈∂Ω, where μ≥−N24 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of Lμu=0 in Ω with boundary data ν+kδ0, where ν is a Radon measure on ∂Ω∖{0}, k∈R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of Lμu=0 in Ω.