On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates

Abstract

We consider the Schrödinger type operator $\mathcal{A}=(1+|x{|}^{\alpha })\Delta -|x{|}^{\beta }$, for $\alpha \in [0,2]$ and $\beta \ge 0$. We prove that, for any $p\in (1,\infty )$, the minimal realization of operator $\mathcal{A}$ in $L^p({\mathbb{R}}^N)$ generates a strongly continuous analytic semigroup $(T_p(t){)}_{t\ge 0}$. For $\alpha \in [0,2)$ and $\beta \ge 2$, we then prove some upper estimates for the heat kernel $k$ associated to the semigroup $(T_p(t){)}_{t\ge 0}$. As a consequence we obtain an estimate for large $|x|$ of the eigenfunctions of $\mathcal{A}$. Finally, we extend such estimates to a class of divergence type elliptic operators.