Schrödinger operators with Leray-Hardy potential singular on the boundary (Q2180546)

Let \({\mathcal L}_\mu\) be the Schrödinger operator defined by \({\mathcal L}_\mu u=-{\Delta} u+\frac{\mu}{|x|^2}u\), in a bounded smooth domain \({\Omega}\subset{\mathbb R}^N_+\), such that \(0\in\partial\Omega\), and where \(\mu\geq-{N^2}/{4}\) is a constant. The authors study the kernel function of the operator \({\mathcal L}_\mu\), showing the existence of a Poisson kernel vanishing at \(0\) and a singular kernel with a singularity at \(0\). For a Radon measure \(\nu\) on \(\partial {\Omega}\setminus\{0\}\) and \(k\in{\mathbb R}\), the existence and uniqueness of weak solutions of \({\mathcal L}_\mu u=0\) in \({\Omega}\) with boundary data \(\nu+k\delta_0\) is obtained. Key steps in the work are the introduction of a distributional identity of certain \({\mathcal L}_\mu\)-harmonic functions in \({\mathbb R}^{N}_+\) and the construction of Kato-type inequalities to build the Poisson kernel.