Vol. 13, No. 5, 2020

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Uniform Sobolev estimates for Schrödinger operators with scaling-critical potentials and applications

Haruya Mizutani

Vol. 13 (2020), No. 5, 1333–1369
Abstract

We prove uniform Sobolev estimates for the resolvent of Schrödinger operators with large scaling-critical potentials without any repulsive condition. As applications, global-in-time Strichartz estimates including some nonadmissible retarded estimates, a Hörmander-type spectral multiplier theorem, and Keller-type eigenvalue bounds with complex-valued potentials are also obtained.

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Keywords
uniform Sobolev estimate, limiting absorption principle, Strichartz estimate, spectral multiplier theorem, eigenvalue bounds, Schrödinger equation
Mathematical Subject Classification 2010
Primary: 35P25, 35J10
Secondary: 35P15, 35Q41
Milestones
Received: 23 October 2017
Revised: 22 September 2018
Accepted: 31 May 2019
Published: 27 July 2020
Authors
Haruya Mizutani
Department of Mathematics
Graduate School of Science
Osaka University
Osaka
Japan