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# Journal of Applied Analysis

Editor-in-Chief: Fechner, Włodzimierz / Ciesielski, Krzysztof

Managing Editor: Gajek, Leslaw

CiteScore 2018: 0.45

SCImago Journal Rank (SJR) 2018: 0.181
Source Normalized Impact per Paper (SNIP) 2018: 0.845

Mathematical Citation Quotient (MCQ) 2018: 0.20

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ISSN
1869-6082
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Volume 20, Issue 1

# On complex Fermi curves of two-dimensional, periodic Schrödinger operators

Alexander Klauer
Published Online: 2014-05-09 | DOI: https://doi.org/10.1515/jaa-2014-0006

## Abstract.

The complex Bloch varieties and the associated Fermi curves of two-dimensional periodic Schrödinger operators with quasi-periodic boundary conditions are defined as complex analytic varieties, the Schrödinger potentials being from the Lorentz–Fourier space $ℱ{\ell }^{\infty ,1}$. Then, an asymptotic analysis of the Fermi curves is performed. The decomposition of a Fermi curve into a compact part, an asymptotically free part, and thin handles, is recovered as expected. Furthermore, it is shown that the set of potentials whose associated Fermi curve has finite geometric genus is a dense subset of $ℱ{\ell }^{\infty ,1}$. Moreover, the Fourier transforms of the potentials are locally isomorphic to perturbed Fourier transforms induced by the handles. Finally, an asymptotic family of parameters describing the sizes of the handles is introduced. These parameters are good candidates for describing parts of the space of all Fermi curves.

MSC: 35P20; 81U40

Revised: 2013-08-26

Accepted: 2013-09-08

Published Online: 2014-05-09

Published in Print: 2014-06-01

Funding Source: DFG

Award identifier / Grant number: SCHM 2395

Citation Information: Journal of Applied Analysis, Volume 20, Issue 1, Pages 55–76, ISSN (Online) 1869-6082, ISSN (Print) 1425-6908,

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© 2014 by Walter de Gruyter Berlin/Boston.