Heat Kernel Estimates for Schr\"odinger Operators with Decay at Infinity on Parabolic Manifolds

TL;DR

This paper derives heat kernel estimates for Schr"odinger operators with decaying potentials on parabolic manifolds, extending known results and providing bounds that enhance understanding of their behavior at infinity.

Contribution

It provides new heat kernel bounds for Schr"odinger operators with decay potentials on parabolic manifolds, complementing existing Euclidean results.

Findings

Matching upper and lower heat kernel bounds established.

Results apply to a broad class of parabolic weighted manifolds.

Enhances understanding of Schr"odinger operators with decaying potentials.

Abstract

We give estimates for positive solutions for the Schr\"odinger equation $(\Delta_\mu+W)u=0$ on a wide class of parabolic weighted manifolds $(M, d\mu)$ when $W$ decays to zero at infinity faster than quadratically. These can be combined with results of Grigor'yan to give matching upper and lower bounds for the heat kernel of the corresponding Schr\"odinger operator $\Delta_\mu+W$. In particular, this appears to complement known results for Schr\"odinger operators on $\mathbf{R}^2$.