Intrinsic ultracontractivity for Schr\"odinger semigroups based on cylindrical fractional Laplacian on the plane

We study Schr\"odinger operators on R² H = (-∂²/∂ x₁²)α/2 + (-∂²/∂ x₂²)α/2 + V, for α ∈ (0,2) and some sufficiently regular, radial, confining potentials V. We obtain necessary and sufficient conditions on intrinsic ultracontractivity for semigroups -tH: \, t ≥ 0\. We also get sharp estimates of first eigenfunctions of H.