The heat kernel of a Schr\"odinger operator with inverse square potential

TL;DR

This paper analyzes the heat kernel of a Schrödinger operator with an inverse square potential, providing precise behavior of harmonic functions and bounds for the heat kernel under general conditions.

Contribution

It offers new detailed descriptions of harmonic functions and establishes upper and lower bounds for the heat kernel of Schrödinger operators with singular potentials.

Findings

Explicit behavior of positive harmonic functions near singularities

Upper bounds for the heat kernel with general conditions

Lower bounds when the harmonic function is an A2 weight

Abstract

We consider the Schr{\"o}dinger operator H = --$\Delta$ + V (|x|) with radial potential V which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of H and give their precise behavior. Second, under quite general conditions we prove an upper bound for the correspond heat kernel p(x, y, t) of the type 0 \textless{} p(x, y, t) $\le$ C t -- N 2 U (min{|x|, $\sqrt$ t})U (min{|y|, $\sqrt$ t}) U ($\sqrt$ t) 2 exp -- |x -- y| 2 Ct for all x, y $\in$ R N and t \textgreater{} 0, where U is a positive harmonic function of H. Third, if U 2 is an A 2 weight on R N , then we prove a lower bound of a similar type.