Scattering theory for repulsive Schrödinger operators and applications to the limit circle problem

Taira, Kouichi

doi:10.2140/apde.2021.14.2101

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Scattering theory for repulsive Schrödinger operators and applications to the limit circle problem

Kouichi Taira

Vol. 14 (2021), No. 7, 2101–2122

DOI: 10.2140/apde.2021.14.2101

Abstract

We study existence of the outgoing/incoming resolvents of repulsive Schrödinger operators which may not be essentially self-adjoint on the Schwartz space. As a consequence, we construct $L^{2}$ -eigenfunctions associated with complex eigenvalues by a standard technique of scattering theory. In particular, we give another proof of the classical result via microlocal analysis: the repulsive Schrödinger operators with large repulsive exponent are not essentially self-adjoint on the Schwartz space.

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Keywords

self-adjointness, microlocal analysis, scattering theory

Mathematical Subject Classification 2010

Primary: 35P25, 81Q10

Milestones

Received: 15 June 2019

Revised: 30 March 2020

Accepted: 31 July 2020

Published: 10 November 2021

Authors

Kouichi Taira
Research Organization of Science and Technology Ritsumeikan University Kusatsu Japan

Vol. 14, No. 7, 2021