Intrinsic ultracontractivity for Schrodinger operators based on fractional Laplacians

Abstract

We study the Feynman-Kac semigroup generated by the Schr{\"o}dinger operator based on the fractional Laplacian $-(-\Delta)^{\alpha/2} - q$ in $\Rd$, for $q \ge 0$, $\alpha \in (0,2)$. We obtain sharp estimates of the first eigenfunction $\phi_1$ of the Schr{\"o}dinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials $q$ such that $\lim_{|x| \to \infty} q(x) = \infty$ and comparable on unit balls we obtain that $\phi_1(x)$ is comparable to $(|x| + 1)^{-d - \alpha} (q(x) + 1)^{-1}$ and intrinsic ultracontractivity holds iff $\lim_{|x| \to \infty} q(x)/\log|x| = \infty$. Proofs are based on uniform estimates of $q$-harmonic functions.