Intrinsic ultracontractivity for Schrodinger operators based on fractional Laplacians

Abstract

We study the Feynman-Kac semigroup generated by the Schr{ö}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 ${\varphi }_1$ of the Schr{ö}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 ${\varphi }_1(x)$ 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.

References in corpus (1)

• Heat kernel of fractional Laplacian in cones