Eigenvalue bounds for Schrödinger operators with complex potentials. II

• Published in: Journal of Spectral Theory
• Main Authors: Frank, Rupert L., Simon, Barry
• Format: Article in Journal/Newspaper
• Language: English
• Published: Ludwig-Maximilians-Universität München 2017
• Subjects: Mathematik; ddc:510; laptev
• Online Access: https://epub.ub.uni-muenchen.de/59575/1/JST-2017-007-003-01.pdf; https://epub.ub.uni-muenchen.de/59575/; http://nbn-resolving.de/urn:nbn:de:bvb:19-epub-59575-2; https://doi.org/10.4171/JST/173

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator −Δ+Vin L2(Rν) with complex potential has absolute value at most a constant times ∥V∥(γ+ν/2)/γγ+ν/2 for 0<γ≤ν/2 in dimension ν≥2. We prove this conjecture for radial potentials if 0<γ<ν/2 and we 'almost disprove' it for general potentials if 1/2<γ<ν/2. In addition, we prove various bounds that hold, in particular, for positive eigenvalues.