Spectral Methods in Surface Superconductivity

ISBN-13: 9780817647964 / Angielski / Twarda / 2010 / 324 str.

Sa Ren Fournais; Bernard Helffer

Opis

In the past decade, the mathematics of superconductivity has been the subject of intense study. This book examines in detail the nonlinear Ginzburg-Landau (GL) functional, the model most commonly used. Specifically, cases in the presence of a strong magnetic field and with a sufficiently large GL parameter kappa are covered. Key topics and features: *Provides a concrete introduction to techniques in spectral theory and PDEs *Offers a complete analysis of the two-dimensional GL-functional with large kappa in the presence of a magnetic field *Treats the three-dimensional case thoroughly *Includes exercises and open problems Spectral Methods in Surface Superconductivity is intended for students and researchers with a graduate level understanding of functional analysis, spectral theory, and PDE analysis. Anything which is not standard is recalled as well as important semiclassical techniques in spectral theory that are involved in the nonlinear study of superconductivity.

Recenzje

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)

Spis treści

Preface.- Notation.- Part I Linear Analysis.- 1 Spectral Analysis of Schr¨odinger Operators.- 2 Diamagnetism.- 3 Models in One Dimension.- 4 Constant Field Models in Dimension 2: Noncompact Case.- 5 Constant Field Models in Dimension 2: Discs and Their Complements.- 6 Models in Dimension 3: R3 or R3,+.- 7 Introduction to Semiclassical Methods for the Schr¨odinger Operator with a Large Electric Potential.- 8 Large Field Asymptotics of the Magnetic Schr¨odinger Operator: The Case of Dimension 2.- 9 Main Results for Large Magnetic Fields in Dimension 3.- Part II Nonlinear Analysis.-10 The Ginzburg–Landau Functional.- 11 Optimal Elliptic Estimates.- 12 Decay Estimates.- 13 On the Third Critical Field HC3.- 14 Between HC2 and HC3 in Two Dimensions.- 15 On the Problems with Corners.- 16 On Other Models in Superconductivity and Open Problems.- A Min-Max Principle.- B Essential Spectrum and Persson’s Theorem.- C Analytic Perturbation Theory.- D About the Curl-Div System.- E Regularity Theorems and Precise Estimates in Elliptic PDE.- F Boundary Coordinates.- References.- Index.

Blurb

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.