Pointwise estimates for the ground states of singular Dirichlet fractional Laplacian

TL;DR

This paper derives precise pointwise bounds for the ground states of certain singular fractional Schrödinger operators on bounded domains, and analyzes their heat kernel behavior and ultracontractivity properties.

Contribution

It provides new sharp estimates for ground states of singular fractional Laplacians with Hardy-type potentials and discusses their heat kernel asymptotics and ultracontractivity.

Findings

Established sharp pointwise estimates for ground states.

Analyzed the intrinsic ultracontractivity property.

Derived sharp large-time asymptotics for heat kernels.

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 $(-\Delta)^{\alpha/2}|_\Om-c|x|^{-\alpha}$, where $(-\Delta)^{\alpha/2}|_\Om$ is the fraction-Laplacien on an open subset $\Om$ in $\R$ with zero exterior condition and $0<c\leq(\frac{d-\alpha}{2})^2$. The intrinsic ultracontractivity property for such operators is discussed as well and a sharp large time asymptotic for their heat kernels is derived.