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

We study Schrödinger operators on ${\mathbb{R}}^2$ $$H={(-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_1^2})}^{\alpha \mathrm{/}2}+{(-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_2^2})}^{\alpha \mathrm{/}2}+V,$$ 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 $H$.