Low lying spectral gaps induced by slowly varying magnetic fields
| Autor: | Bernard Helffer, Horia D. Cornean, Radu Purice |
|---|---|
| Přispěvatelé: | Department of Mathematical Sciences [Aalborg], Aalborg University [Denmark] (AAU), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Structure (category theory)
FOS: Physical sciences 01 natural sciences Mathematics - Spectral Theory Matrix (mathematics) symbols.namesake Quantum mechanics 0103 physical sciences FOS: Mathematics [MATH]Mathematics [math] 0101 mathematics Spectral Theory (math.SP) Eigenvalues and eigenvectors Mathematical Physics Mathematics Operator (physics) 010102 general mathematics Mathematical analysis Spectrum (functional analysis) Mathematical Physics (math-ph) Magnetic field symbols 010307 mathematical physics Analysis Intensity (heat transfer) Schrödinger's cat |
| Zdroj: | Cornean, D H, Helffer, B & Purice, R 2017, ' Low lying spectral gaps induced by slowly varying magnetic fields ', Journal of Functional Analysis, vol. 273, no. 1, pp. 206-282 . https://doi.org/10.1016/j.jfa.2017.04.002 Journal of Functional Analysis Journal of Functional Analysis, Elsevier, 2017, 273, pp.206--282 |
| ISSN: | 0022-1236 1096-0783 |
| DOI: | 10.1016/j.jfa.2017.04.002 |
| Popis: | We consider a periodic Schr\"odinger 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 Schr\"odinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure. First, we construct an effective Hofstadter-like 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 do not cross but their ranges might overlap. The crossing case is more difficult and will be considered elsewhere. Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted to slowly varying magnetic fields. As an application, we prove in the overlapping case the appearance of spectral islands separated by gaps. Comment: 55 pages, to appear in J. Funct. Anal |
| Databáze: | OpenAIRE |
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