Effet d'Aharonov-Bohm sur un état borné de l'équation de Schrödinger. (Aharonov-Bohm effect for a bounded state of the Schrödinger equation)
type: Item
instance of: scholarly article
title: Effet d'Aharonov-Bohm sur un état borné de l'équation de Schrödinger. (Aharonov-Bohm effect for a bounded state of the Schrödinger equation)
zbMATH Open document ID: 0702.35186
DOI: 10.1007/BF01217743
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: The author discusses the groundstate energy splitting of a Schrödinger operator when an Aharanov-Bohm magnetic field is turned on (Schrödinger operator with vector potential whose magnetic field vanishes outside a certain disc in \(R^ 2)\). The splitting turns out to be exponentially small in the Planck constant (quasiclassical limit). The decay rate is given as the minimal length - measured in the Agmon metric - of paths sourrounding the region with magnetic field. Furthermore the result is extended to construct a counterexample to the so called paramagnetic conjecture of Hogreve, Schrader, and Seiler. The proof of these results uses the decoupling method of wells as developed by the author and \textit{J. Sjöstrand} [Commun. Partial Differ. Equations 9, 337-408 (1984; Zbl 0546.35053) and Ann. Inst. Henri Poincaré, Phys. Théor. 42, 127-212 (1985; Zbl 0595.35031)] as an essentiel tool.
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 4150923
zbMATH Keywords: energy splitting
zbMATH Keywords: Aharanov-Bohm magnetic field
zbMATH Keywords: quasiclassical limit
zbMATH Keywords: decay rate
zbMATH Keywords: Agmon metric
zbMATH Keywords: paramagnetic conjecture
author: Bernard Helffer
reviewed by: Heinz Siedentop
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4150923