General Kernel estimates of Schrödinger type operators with unbounded diffusion terms

Abstract

We prove first that the realization $A_{\min }$ of $A:=\mathrm{div}(Q\nabla )-V$ in $L^2({\mathbb{R}}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2({\mathbb{R}}^d)$ which coincides on $L^2({\mathbb{R}}^d)\cap C_b({\mathbb{R}}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b({\mathbb{R}}^d)$. Moreover, using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\min }$ in the case of polynomially and exponentially diffusion and potential coefficients.

21 pages, no figures