Spectral Functions in Mathematics and Physics: 1st Edition (Hardback) book cover

Spectral Functions in Mathematics and Physics

1st Edition

By Klaus Kirsten

Chapman and Hall/CRC

400 pages | 16 B/W Illus.

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pub: 2001-12-13
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Description

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.

Reviews

"Spectral geometry and spectral analysis play an important role not only in global analysis but also in certain other areas of mathematics and physics. The book Spectral Functions in Mathematics and Physics is suitable for a wide audience of both experts and non experts in these fields--it is a well-written introduction to the field by one of the experts … [it] will be useful to both mathematicians and mathematical physicists."

- SIAM Review, 2003

Table of Contents

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

Subject Categories

BISAC Subject Codes/Headings:
MAT012000
MATHEMATICS / Geometry / General
SCI040000
SCIENCE / Mathematical Physics
SCI055000
SCIENCE / Physics