General Kernel estimates of Schr\"odinger type operators with unbounded diffusion terms

TL;DR

This paper establishes kernel estimates and spectral properties for Schrödinger-type operators with unbounded diffusion terms, demonstrating ultracontractivity and symmetry of the generated semigroup in $L^2$ spaces.

Contribution

It introduces new kernel bounds for Schrödinger operators with unbounded coefficients using Lyapunov functions, extending understanding of their spectral and semigroup properties.

Findings

Proves the semigroup generated by the operator is symmetric, sub-Markovian, and ultracontractive.

Derives pointwise upper bounds for the heat kernel using Lyapunov functions.

Analyzes spectral properties under polynomial and exponential coefficient growth.

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.