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

TL;DR

This paper studies a class of Schr"odinger operators with unbounded coefficients, proving generation of analytic semigroups in L^p spaces, deriving heat kernel estimates, and analyzing eigenfunction behavior.

Contribution

It establishes the generation of strongly continuous analytic semigroups for these operators and provides new heat kernel and eigenfunction estimates, extending to divergence form elliptic operators.

Findings

Generation of analytic semigroups in L^p spaces for the operators.

Upper bounds for heat kernels associated with the operators.

Eigenfunction estimates for large |x| extending to divergence form operators.

Abstract

We consider the Schr\"odinger 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 $L^p(\R^N)$ 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.