Intrinsic ultracontractivity for fractional Schr\"odinger operators

TL;DR

This paper derives precise estimates for ground states and heat kernels of fractional Schr"odinger operators with singular potentials, establishing intrinsic ultracontractivity and large-time asymptotics on bounded domains.

Contribution

It provides sharp pointwise bounds for ground states and heat kernels of fractional Schr"odinger operators with singular potentials, and analyzes their ultracontractivity properties.

Findings

Sharp ground state estimates for fractional Schr"odinger operators.

Large time asymptotics for heat kernels.

Intrinsic ultracontractivity established for these operators.

Abstract

We establish sharp pointwise estimates for the ground states of some singular fractional Schr\"odinger operators on relatively compact Euclidean subsets. The considered operators are of the type $(-\Delta)^{\alpha/2}|_\Omega-V$, where $V\in L^1_{Loc}$ and $(-\Delta)^{\alpha/2}|_\Omega$ is the fractional-Laplacian on an open subset $\Omega$ in $\mathbb{R}^d$ with zero exterior condition . The intrinsic ultracontractivity property for such operators is discussed as well and a sharp large time asymptotic for their heat kernels is derived.