Schr\"odinger equations with smooth measure potential and general measure data

TL;DR

This paper investigates Schr"odinger equations with singular measure potentials and general measure data, establishing conditions for solutions and analyzing their regularity and stability.

Contribution

It introduces a comprehensive framework for Schr"odinger equations with measure potentials and general measure data, including existence criteria and regularity results.

Findings

Established necessary and sufficient conditions for solution existence

Proved regularity and stability of solutions

Covered broad class of Dirichlet operators, including integro-differential operators

Abstract

We study equations driven by Schr\"odinger operators consisting of a self-adjoint Dirichlet operator and a singular potential, which belongs to a class of positive Borel measures absolutely continuous with respect to a capacity generated by the operator. In particular, we cover positive potentials exploding on a set of capacity zero. The right-hand side of equations is allowed to be a general bounded Borel measure. The class of self-adjoint Dirichlet operators is quite large. Examples include integro-differential operators with the local part of divergence form. We give a necessary and sufficient condition for the existence of a solution, and prove some regularity and stability results.