  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 $H$.
