Intrinsic ultracontractivity for Schrödinger semigroups based on cylindrical fractional Laplacian on the plane
Tadeusz Kulczycki, Kinga Sztonyk · Semigroup Forum · 2025
We study Schrödinger 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.