Schr\"odinger Operators With $A_\infty$ Potentials

TL;DR

This paper establishes pointwise upper bounds for the heat kernel of Schr"odinger operators with $A_ abla$ potentials, providing explicit formulas for quadratic potentials and bounds for potentials in reverse Hölder classes.

Contribution

It introduces new bounds for the heat kernel of Schr"odinger operators with $A_ abla$ potentials and explicitly computes the kernel for quadratic potentials.

Findings

Established pointwise upper bounds for the heat kernel with $A_ abla$ potentials.

Derived lower bounds for potentials in reverse Hölder classes.

Explicitly computed the heat kernel for quadratic polynomial potentials.

Abstract

We study the heat kernel $p(x,y,t)$ associated to the real Schr\"odinger operator $H = -\Delta + V$ on $L^2(\mathbb{R}^n)$, $n \geq 1$. Our main result is a pointwise upper bound on $p$ when the potential $V \in A_\infty$. In the case that $V\in RH_\infty$, we also prove a lower bound. Additionally, we compute $p$ explicitly when $V$ is a quadratic polynomial.