Spectral Functions in Mathematics and Physics

Spectral Functions in Mathematics and Physics

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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.show more

Product details

  • Hardback | 400 pages
  • 162.1 x 240.3 x 26.9mm | 721.22g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 16 black & white illustrations
  • 158488259X
  • 9781584882596

Review quote

"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, 2003show more

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 sectionshow more

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