On confining potentials and essential self-adjointness for Schr\"odinger operators on bounded domains in R^n

TL;DR

This paper establishes optimal growth conditions on the potential near the boundary of a bounded domain in R^n that ensure the Schrödinger operator is essentially self-adjoint, refining known criteria with sharp logarithmic corrections.

Contribution

It introduces the weakest growth conditions, including optimal logarithmic corrections, guaranteeing essential self-adjointness of Schrödinger operators on bounded domains.

Findings

Optimal logarithmic growth conditions for potential near boundary

Sharp constants in growth conditions are proven to be optimal

Refined Agmon estimates and Hardy inequalities underpin the results

Abstract

Let $\Omega$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-dimension 1, and let $H=-\Delta +V(x)$ be a Schr\"odinger operator on $\Omega$ with potential V locally bounded. We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on $C_0^\infty(\Omega)$. As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition $V(x)\geq \frac{3}{4d(x)^2}$, where $d(x)=dist(x,\partial\Omega)$. The constant 1 in front of each logarithmic term in Theorem 2 is optimal. The proof is based on a refined Agmon exponential estimate combined with a well known multidimensional Hardy inequality.