Effet d'Aharonov-Bohm sur un état borné de l'équation de Schrödinger. (Aharonov-Bohm effect for a bounded state of the Schrödinger equation)
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.
scientific article; zbMATH DE number 4150923
scientific article