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.
Key topics and features of the work:
* Provides a concrete introduction to techniques in spectral theory and partial differential equations
* Offers a complete analysis of the two-dimensional Ginzburg–Landau functional with large kappa in the presence of a magnetic field
* Treats the three-dimensional case thoroughly
* Includes open problems
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.
Author(s): Søren Fournais, Bernard Helffer (auth.)
Series: Progress in Nonlinear Differential Equations and Their Applications 77
Edition: 1
Publisher: Birkhäuser Basel
Year: 2009
Language: English
Pages: 324
Tags: Functional Analysis; Strongly Correlated Systems, Superconductivity; Partial Differential Equations; Special Functions
Front Matter....Pages I-XX
Front Matter....Pages 1-1
Spectral Analysis of Schrödinger Operators....Pages 3-17
Diamagnetism....Pages 19-30
Models in One Dimension....Pages 31-43
Constant Field Models in Dimension 2: Noncompact Case....Pages 45-50
Constant Field Models in Dimension 2: Discs and Their Complements....Pages 51-65
Models in Dimension 3: $$\mathbb{R}^3$$ or $$\mathbb{R}^{3,+}$$ ....Pages 67-77
Introduction to Semiclassical Methods for the Schrödinger Operator with a Large Electric Potential....Pages 79-94
Large Field Asymptotics of the Magnetic Schrödinger Operator: The Case of Dimension 2....Pages 95-115
Main Results for Large Magnetic Fields in Dimension 3....Pages 117-137
Front Matter....Pages 140-140
The Ginzburg–Landau Functional....Pages 141-156
Optimal Elliptic Estimates....Pages 157-178
Decay Estimates....Pages 179-192
On the Third Critical Field $$H_{C_3}$$ ....Pages 193-207
Between $$H_{C_2}$$ and $$H_{C_3}$$ in Two Dimensions....Pages 209-238
On the Problems with Corners....Pages 239-260
On Other Models in Superconductivity and Open Problems....Pages 261-268
Back Matter....Pages 269-324
