Kernel estimates for a class of Schrödinger semigroups

We prove short time estimates for the heat kernels of certain Schrödinger operators with unbounded potentials in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{R}}^N$$ \end{document}. The asymptotic distribution of the eigenvalues is also considered.