Band functions in presence of magnetic steps

Peter D. Hislop (1) , Nicolas Popoff (2) , Nicolas Raymond (3) , Mikael Sundqvist (4)

1 University of Kentucky
2 Équipe EDP et Physique Mathématique
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 Department of Mathematics [Lund University]

We complete the analysis of the band functions for two-dimensional magnetic Schrödinger operators with piecewise constant magnetic fields. The discontinuity of the magnetic field can create edge currents that flow along the discontinuity that have been described by physicists. Properties of these edge currents are directly related to the behavior of the band functions. The effective potential of the fiber operator is an asymmetric double well (eventually degenerated) and the analysis of the splitting of the bands incorporates the asymmetry. If the magnetic field vanishes, the reduced operator has essential spectrum and we provide an explicit description of the band functions located below the essential spectrum. For non degenerate magnetic steps, we provide an asymptotic expansion of the band functions at infinity. We prove that when the ratio of the two magnetic fields is rational, a splitting of the band functions occurs and has a natural order, predicted by numerical computations.

Mots clés: magnetic Schrödinger operators; edge currents; band functions

Domaines: Mathématiques [math]

Peter D. Hislop, Nicolas Popoff, Nicolas Raymond, Mikael Sundqvist. Band functions in presence of magnetic steps. Mathematical Models and Methods in Applied Sciences, 2016, 26 (1), pp.161-184. ⟨10.1142/S0218202516500056⟩. ⟨hal-01102553⟩