Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms

Abstract

We prove that the heat kernel associated to the Schrödinger type operator $A:=(1+|x{|}^{\alpha })\Delta -|x{|}^{\beta }$ satisfies the estimate

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 $N>2,\alpha \ge 2$ and $\beta >\alpha -2$. We also obtain an estimate of the eigenfunctions of $A$.