Intrinsic ultracontractivity for Schrödinger semigroups based on cylindrical fractional Laplacian on the plane

Abstract

We study Schrödinger operators on ${\mathbb{R}}^2$ $$H={\left(-\frac{{\partial }^2}{\partial x_1^2}\right)}^{\hat{I}\pm /2}+{\left(-\frac{{\partial }^2}{\partial x_2^2}\right)}^{\hat{I}\pm /2}+V,$$ for $\hat{I}\pm \in (0,2)$ 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 $H$.