Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians
By K. Kaleta, Tadeusz Kulczycki. 2009
48 citations recorded by the source corpus.
We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian − ( − Δ)α/2 − q in Rd, 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.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d8801fbe56df60ee7f7ecbb189b648e08b427a31
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In Computational Theory and Mathematics
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Citations in the corpus
- Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of L\'{e}vy processes (2012, background): “More recently, IUC has been investigated also in the case of semigroups generated by fractional Laplacians, and fractional and relativistic Schrödinger operators [26, 27, 46, 47, 55, 57, 58], as well as for more general symmetric [37] and nonsymmetric [49] Lévy processes in bounded domains.”