Semiclassical resolvent bounds for compactly supported radial potentials

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator −h2Δ+V(|x|)−E in dimension n≥2, where h,E>0, and V:[0,∞)→R is L∞ and compactly supported. The weighted resolvent norm grows no faster than exp⁡(Ch−1), while an exterior weighted norm...

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Bibliographic Details
Published in:Journal of functional analysis 2023-04, Vol.284 (7), p.109835, Article 109835
Main Authors: Datchev, Kiril, Galkowski, Jeffrey, Shapiro, Jacob
Format: Article
Language:English
Subjects:
Radial potential
Resolvent estimate
Schrödinger operator
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Summary:We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator −h2Δ+V(|x|)−E in dimension n≥2, where h,E>0, and V:[0,∞)→R is L∞ and compactly supported. The weighted resolvent norm grows no faster than exp⁡(Ch−1), while an exterior weighted norm grows ∼h−1. We introduce a new method based on the Mellin transform to handle the two-dimensional case.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2022.109835