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 $$k(t,x,y)\le c_1{e}^{{\lambda }_{0}t}{e}^{c_2{t}^{-b}}\frac{(|x||y|{)}^{-\frac{N-1}{2}-\frac{\beta -\alpha }{4}}}{1+|y{|}^{\alpha }}{e}^{-\frac{2}{\beta -\alpha +2}|x{|}^{\frac{\beta -\alpha +2}{2}}}{e}^{-\frac{2}{\beta -\alpha +2}|y{|}^{\frac{\beta -\alpha +2}{2}}}$$ 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 $N>2,\alpha \ge 2$ and $\beta >\alpha -2$. We also obtain an estimate of the eigenfunctions of $A$.