Semiclassical reduction for magnetic Schrödinger operator with periodic zero-range potentials and applications

Article type: Research Article

Authors: Helffer, Bernard | Pankrashkin, Konstantin^;

Affiliations: Département de Mathématiques, Université Paris Sud, Bâtiment 425, 91405 Orsay Cedex, France | Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany

Note: [] Corresponding author: Konstantin Pankrashkin, Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany. Tel.: +49 30 2093 2352; Fax: +49 30 2093 2727; E-mail: [email protected].

Abstract: The two-dimensional Schrödinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields and a weak coupling we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like operators. This shows the existence of parts of Cantor structure in the spectrum for special values of the magnetic flux.

Keywords: Schrödinger operator, periodic perturbation, Cantor spectrum, magnetic field, zero-range potential

DOI: 10.3233/ASY-2008-0923

Journal: Asymptotic Analysis, vol. 63, no. 1-2, pp. 1-27, 2009