E-Book, Englisch, 400 Seiten
Kirsten Spectral Functions in Mathematics and Physics
Erscheinungsjahr 2010
ISBN: 978-1-4200-3546-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
E-Book, Englisch, 400 Seiten
ISBN: 978-1-4200-3546-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
The literature on the spectral analysis of second order elliptic differential operators contains a great deal of information on the spectral functions for explicitly known spectra. The same is not true, however, for situations where the spectra are not explicitly known. Over the last several years, the author and his colleagues have developed new, innovative methods for the exact analysis of a variety of spectral functions occurring in spectral geometry and under external conditions in statistical mechanics and quantum field theory.
Spectral Functions in Mathematics and Physics presents a detailed overview of these advances. The author develops and applies methods for analyzing determinants arising when the external conditions originate from the Casimir effect, dielectric media, scalar backgrounds, and magnetic backgrounds. The zeta function underlies all of these techniques, and the book begins by deriving its basic properties and relations to the spectral functions. The author then uses those relations to develop and apply methods for calculating heat kernel coefficients, functional determinants, and Casimir energies. He also explores applications in the non-relativistic context, in particular applying the techniques to the Bose-Einstein condensation of an ideal Bose gas.
Self-contained and clearly written, Spectral Functions in Mathematics and Physics offers a unique opportunity to acquire valuable new techniques, use them in a variety of applications, and be inspired to make further advances.
Zielgruppe
Graduate students and researchers in physics, global analysis, geometry, and topology
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
INTRODUCTION
A FIRST LOOK AT ZETA FUNCTIONS AND HEAT TRACES
Zeta Function in Quantum Field Theory
Statistical Mechanics of Finite Systems: Bose-Einstein Condensation
Local versus Global Boundary Conditions
ZETA FUNCTIONS ON GENERALIZED CONES AND RELATED MANIFOLDS
Scalar Field on the Three-Dimensional Ball
Scalar Field on the D-Dimensional Generalized Cone
Spinor Field with Global and Local Boundary Conditions
Forms with Absolute and Relative Boundary Conditions
Oblique Boundary Conditions on the Generalized Cone
Further Examples on a Related Geometry
CALCULATION OF HEAT KERNEL COEFFCIENTS VIA SPECIAL CASES
Heat Equation Asymptotics for Manifolds without Boundary
General Form for Dirichlet and Robin Boundary Conditions
Heat Kernel Coefficients on the Generalized Cone
Determination of the General Heat Kernel Coefficients
Mixed Boundary Conditions
Special Case Calculations for Mixed Boundary Conditions
Determination of the Mixed Heat Kernel Coefficients
Oblique Boundary Conditions
Leading Heat Equation Asymptotics with Spectral Boundary Conditions
Summary of the Results
Further Boundary Conditions
HEAT CONTENT ASYMPTOTICS
General Form of the Heat Content Coefficients
Dirichlet Boundary Conditions
Robin Boundary Conditions
Heat Content Asymptotics on the Generalized Cone
Mixed Boundary Conditions
FUNCTIONAL DETERMINANTS
Some One-Dimensional Examples
Scalar Field
Spinor Field with Global and Local Boundary Conditions
Forms with Absolute and Relative Boundary Conditions
Determinants by Conformal Transformation
CASIMIR ENERGIES
Scalar Field
Spinor Field with Global and Local Boundary Conditions
Electromagnetic Field with and without Medium
Massive Scalar Field
Massive Spinor Field with Local Boundary Conditions
GROUND STATE ENERGIES UNDER THE INFLUENCE OF EXTERNAL FIELDS
Formalism: Scattering Theory and Ground State Energy
Examples and General Results
Spinor Field in the Background of a Finite Radius Flux Tube
BOSE-EINSTEIN CONDENSATION OF IDEAL BOSE GASES UNDER EXTERNAL CONDITIONS
Ideal Bose Gases in the Grand Canonical Description
Canonical Description of Ideal Bose-Einstein Condensates
Microcanonical Condensate Fluctuations
CONCLUSIONS
APPENDICES
Basic Zeta Functions
Conformal Relations between Geometric Tensors
Application of Index Theorems
Representations for the Asymptotic Contributions
Perturbation Theory for the Logarithm of the Jost Function
REFERENCES
INDEX
Each chapter also includes an Introduction and Concluding Remarks section
