Location of zeros of non-trivial positive supersolutions to Schr\"odinger equations

TL;DR

This paper investigates the zero set structure of non-negative supersolutions to Schr"odinger equations with singular potentials, including Levy operators and diffusions, addressing a question by H. Brezis.

Contribution

It characterizes the zero set structure of supersolutions for a broad class of Schr"odinger operators with singular potentials, including Coulomb and harmonic potentials.

Findings

Analyzes the structure of zeros for supersolutions to Schr"odinger equations.

Includes operators like Levy type and divergence form diffusions.

Considers potentials as positive measures, including generalized cases.

Abstract

We study Schr\"odinger operators on $L^2(E;m)$ of the form $-A+V$ with singular potentials $V$. We address the question posed by H. Brezis about the structure of the set $\{u=0\}$ for non-negative supersolutions to $-Au+Vu=0$. The class of operators $A$ we study in the paper includes, in particular, symmetric Levy type operators and symmetric diffusions in divergence form, with strictly positive Green functions. The class of potentials $V$ consists of positive smooth measures, which contains, in particular, Coulomb potentials and harmonic potentials, as well as generalized potentials, i.e. positive Borel measures concentrated on $m$-negligible sets.