Intrinsic ultracontractivity for Schr\"odinger semigroups based on cylindrical fractional Laplacian on the plane

TL;DR

This paper investigates the conditions under which Schr"odinger semigroups generated by fractional Laplacians on the plane exhibit intrinsic ultracontractivity, providing sharp eigenfunction estimates for certain confining potentials.

Contribution

It establishes necessary and sufficient conditions for intrinsic ultracontractivity of Schr"odinger semigroups with cylindrical fractional Laplacians on \\mathbb{R}^2, including eigenfunction estimates.

Findings

Necessary and sufficient conditions for ultracontractivity.

Sharp estimates of first eigenfunctions.

Analysis for fractional Laplacian-based Schrödinger operators.

Abstract

We study Schr\"odinger operators on $\mathbb{R}^2$ $$ H = \left(-\frac{\partial^2}{\partial x_1^2}\right)^{\alpha/2} + \left(-\frac{\partial^2}{\partial x_2^2}\right)^{\alpha/2} + V, $$ for $\alpha \in (0,2)$ and some sufficiently regular, radial, confining potentials $V$. We obtain necessary and sufficient conditions on intrinsic ultracontractivity for semigroups $\{e^{-tH}: \, t \ge 0\}$. We also get sharp estimates of first eigenfunctions of $H$.