Kernel estimates for Schr\"odinger type operators with unbounded diffusion and potential terms

TL;DR

This paper establishes precise heat kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms, providing insights into their spectral properties and eigenfunction behavior.

Contribution

It introduces new heat kernel bounds for a class of Schrödinger operators with unbounded coefficients, extending previous results to more general unbounded settings.

Findings

Derived explicit heat kernel upper bounds for the operator

Provided estimates for eigenfunctions of the operator

Extended analysis to operators with unbounded diffusion and potential terms

Abstract

We prove that the heat kernel associated to the Schr\"odinger type operator $A:=(1+|x|^\alpha)\Delta-|x|^\beta$ satisfies the estimate $$k(t,x,y)\leq c_1e^{\lambda_0t}e^{c_2t^{-b}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{\beta-\alpha}{4}}}{1+|y|^\alpha} e^{-\frac{2}{\beta-\alpha+2}|x|^{\frac{\beta-\alpha+2}{2}}} e^{-\frac{2}{\beta-\alpha+2}|y|^{\frac{\beta-\alpha+2}{2}}} $$ for $t>0,|x|,|y|\ge 1$, where $c_1,c_2$ are positive constants and $b=\frac{\beta-\alpha+2}{\beta+\alpha-2}$ provided that $N>2,\,\alpha\geq 2$ and $\beta>\alpha-2$. We also obtain an estimate of the eigenfunctions of $A$.