Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian −(−∆)−q in R, for q ≥ 0, α ∈ (0, 2). We obtain sharp estimates of the first eigenfunction φ1 of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim|x|→∞ q(x) = ∞ and comparable on unit balls we obtain that φ1(x) is comparable to (|x|+1)−d−α(q(x)+1)−1 and intrinsic ultracontractivity holds iff lim|x|→∞ q(x)/ log |x| =∞. Proofs are based on uniform estimates of q-harmonic functions.