Semiclassical resolvent bounds for short range L∞ potentials with singularities at the origin

Article type: Research Article

Authors: Shapiro, Jacob^{; *}

Affiliations: Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA

Correspondence: [*] Corresponding author. E-mail: jshapiro1@udayton.edu.

Abstract: We consider, for h,E>0, resolvent estimates for the semiclassical Schrödinger operator −h2Δ+V−E. Near infinity, the potential takes the form V=VL+VS, where VL is a long range potential which is Lipschitz with respect to the radial variable, while VS=O(|x|−1(log|x|)−ρ) for some ρ>1. Near the origin, |V| may behave like |x|−β, provided 0⩽β<2(3−1). We find that, for any ρ˜>1, there are C,h0>0 such that we have a resolvent bound of the form exp(Ch−2(log(h−1))1+ρ˜) for all h∈(0,h0]. The h-dependence of the bound improves if VS decays at a faster rate toward infinity.

Keywords: Resolvent estimate, Schrödinger operator, short range potential

DOI: 10.3233/ASY-231872

Journal: Asymptotic Analysis, vol. 136, no. 3-4, pp. 157-180, 2024