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