Intrinsic ultracontractivity for Schrodinger operators based on fractional Laplacians

TL;DR

This paper investigates the spectral properties and ultracontractivity of Schr{"o}dinger operators involving fractional Laplacians, providing sharp eigenfunction estimates and conditions for ultracontractivity based on potential growth.

Contribution

It offers new sharp estimates for the first eigenfunction and characterizes intrinsic ultracontractivity for fractional Schr{"o}dinger operators with unbounded potentials.

Findings

Eigenfunction $_1$ behaves like $(|x|+1)^{-d-{eta}{2}}(q(x)+1)^{-1}$ under certain conditions.

Ultracontractivity holds iff $q(x)/ ext{log}|x| o ty$ as $|x| oty$.

Provides uniform estimates of $q$-harmonic functions.

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.