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

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 $H$.