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

  • Alexander Klauer EMAIL logo
Published/Copyright: May 9, 2014
Journal of Applied Analysis
From the journal Journal of Applied Analysis Volume 20 Issue 1

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 ℱℓ∞,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 ℱℓ∞,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.

Keywords: Periodic Schrödinger operator; Bloch variety; Fermi curve; finite type
MSC: 35P20; 81U40

Funding source: DFG

Award Identifier / Grant number: SCHM 2395

The author thanks Martin U. Schmidt, the advisor of his dissertation project, of which the present paper is an outgrowth, and the referee for their suggestions.

Received: 2013-1-24
Revised: 2013-8-26
Accepted: 2013-9-8
Published Online: 2014-5-9
Published in Print: 2014-6-1

© 2014 by Walter de Gruyter Berlin/Boston

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  1. Frontmatter
  2. Porosity and the bounded linear regularity property
  3. Analytic in planar domains functions with preassigned asymptotic set
  4. Multiplication of the distributions (x±i0)z
  5. Hermite–Hadamard type inequalities for extended s-convex functions on the co-ordinates in a rectangle
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  7. On complex Fermi curves of two-dimensional, periodic Schrödinger operators
  8. On the geometrical properties of some classes of complex harmonic functions defined by analytic or coefficient conditions with complex parameter
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https://doi.org/10.1515/jaa-2014-0006
Keywords for this article
Periodic Schrödinger operator; Bloch variety; Fermi curve; finite type

Articles in the same Issue

  1. Frontmatter
  2. Porosity and the bounded linear regularity property
  3. Analytic in planar domains functions with preassigned asymptotic set
  4. Multiplication of the distributions (x±i0)z
  5. Hermite–Hadamard type inequalities for extended s-convex functions on the co-ordinates in a rectangle
  6. Modes of ideal continuity and the additive property in the Riesz space setting
  7. On complex Fermi curves of two-dimensional, periodic Schrödinger operators
  8. On the geometrical properties of some classes of complex harmonic functions defined by analytic or coefficient conditions with complex parameter
  9. On the order of weighted approximation of unbounded functions by some generalizations of Gadjiev–Ibragimov operators
  10. On series whose rearrangements possess discrete sets of limit points
  11. A system of two conservation laws with flux conditions and small viscosity
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