Low lying spectral gaps induced by slowly varying magnetic fields

Horia D Cornean (1) , Bernard Helffer (2, 3) , Radu Purice (4)

1 Department of Mathematical Sciences [Aalborg]
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 LMJL - Laboratoire de Mathématiques Jean Leray
4 IMAR - "Simion Stoilow" Institute of Mathematics

Consider a periodic Schrödinger operator in two dimensions, perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schrödinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure. First, we construct an effective magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues never cross. The crossing case is more difficult and it will be considered elsewhere. Secondly, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted for slowly varying magnetic fields.

Domaines: Mathématiques [math]

Horia D Cornean, Bernard Helffer, Radu Purice. Low lying spectral gaps induced by slowly varying magnetic fields. Journal of Functional Analysis, 2017, 273, pp.206--282. ⟨hal-01434777⟩