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Spectral Methods in Surface Superconductivity

  • Textbook
  • © 2010

Overview

Authors:
  1. Søren Fournais

    1. Department of Mathematical Sciences, University of Aarhus, Aarhus C, Denmark

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  2. Bernard Helffer

    1. , Département de Mathématiques, Université Paris-Sud and CNRS, Orsay Cedex, France

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    You can also search for this author in PubMed   Google Scholar

  • Covers groundbreaking results in the fast growing field of superconductivity
  • Provides a concrete introduction to PDEs and spectral methods
  • Covers both two- and three-dimensional cases of Ginzburg–Landau function extensively
  • Open problems included
  • Includes supplementary material: sn.pub/extras

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications (PNLDE, volume 77)

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Table of contents (16 chapters)

  1. Front Matter

    Pages I-XX
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  2. Linear Analysis

    1. Front Matter

      Pages 1-1
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    2. Spectral Analysis of Schrödinger Operators

      • Søren Fournais, Bernard Helffer
      Pages 3-17

    3. Diamagnetism

      • Søren Fournais, Bernard Helffer
      Pages 19-30

    4. Models in One Dimension

      • Søren Fournais, Bernard Helffer
      Pages 31-43

    5. Constant Field Models in Dimension 2: Noncompact Case

      • Søren Fournais, Bernard Helffer
      Pages 45-50

    6. Constant Field Models in Dimension 2: Discs and Their Complements

      • Søren Fournais, Bernard Helffer
      Pages 51-65

    7. Models in Dimension 3: \(\mathbb{R}^3\) or \(\mathbb{R}^{3,+}\)

      • Søren Fournais, Bernard Helffer
      Pages 67-77

    8. Introduction to Semiclassical Methods for the Schrödinger Operator with a Large Electric Potential

      • Søren Fournais, Bernard Helffer
      Pages 79-94

    9. Large Field Asymptotics of the Magnetic Schrödinger Operator: The Case of Dimension 2

      • Søren Fournais, Bernard Helffer
      Pages 95-115

    10. Main Results for Large Magnetic Fields in Dimension 3

      • Søren Fournais, Bernard Helffer
      Pages 117-137

  3. Nonlinear Analysis

    1. Front Matter

      Pages 140-140
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    2. The Ginzburg–Landau Functional

      • Søren Fournais, Bernard Helffer
      Pages 141-156

    3. Optimal Elliptic Estimates

      • Søren Fournais, Bernard Helffer
      Pages 157-178

    4. Decay Estimates

      • Søren Fournais, Bernard Helffer
      Pages 179-192

    5. On the Third Critical Field \(H_{C_3}\)

      • Søren Fournais, Bernard Helffer
      Pages 193-207

    6. Between \(H_{C_2}\) and \(H_{C_3}\) in Two Dimensions

      • Søren Fournais, Bernard Helffer
      Pages 209-238

    7. On the Problems with Corners

      • Søren Fournais, Bernard Helffer
      Pages 239-260

    8. On Other Models in Superconductivity and Open Problems

      • Søren Fournais, Bernard Helffer
      Pages 261-268

  4. Back Matter

    Pages 269-324
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Keywords

About this book

During the past decade, the mathematics of superconductivity has been the subject of intense activity. This book examines in detail the nonlinear Ginzburg–Landau functional, the model most commonly used in the study of superconductivity. Specifically covered are cases in the presence of a strong magnetic field and with a sufficiently large Ginzburg–Landau parameter kappa.

Spectral Methods in Surface Superconductivity is intended for students and researchers with a graduate-level understanding of functional analysis, spectral theory, and the analysis of partial differential equations. The book also includes an overview of all nonstandard material as well as important semi-classical techniques in spectral theory that are involved in the nonlinear study of superconductivity.

Reviews

From the reviews:

“The book is concerned with the analysis of mathematical problems connected with the theory of superconductivity. The authors consider a standard basic model of superconductivity described by the Ginzburg-Landau functional. … The authors attempt to make the book self-contained, having graduate students and researchers in mind. For this purpose, at the end of the book they add various appendices containing somewhat standard material. … The book concludes with a fairly complete bibliography on the subject.” (Yuri A. Kordyukov, Mathematical Reviews, Issue 2011 j)

Authors and Affiliations

  • Department of Mathematical Sciences, University of Aarhus, Aarhus C, Denmark

    Søren Fournais

  • , Département de Mathématiques, Université Paris-Sud and CNRS, Orsay Cedex, France

    Bernard Helffer

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