Spectral functions in mathematics and physics (Q2770560)

This book discusses some recent applications of spectral geometry to problems arising from both mathematics and physics. After a brief introductory chapter, in the second chapter the author explains some basic properties of zeta functions and heat traces and their relevance to quantum field theory and statistical mechanics. Also crucial differences between local and global boundary conditions for these spectral functions are discussed. In the third chapter, zeta functions are discussed on generalized cones. It is here, that recently developed methods are explained, which allow for the analysis of spectral functions for cases where no explicit knowledge of eigenvalues is available. The fourth chapter deals with heat kernel coefficients and the fifth chapter with heat content asymptotics. In chapter six, the author turns to global problems by discussing the functional determinant. The remaining chapters deal with topics from mathematics and physics: Casimir energies, ground state energies with external fields, and Bose-Einstein condensation of ideal Bose gases. NEWLINENEWLINENEWLINEThe author is an expert on the application of the so-called `special case' method of computation. Using special functions, one is often able to compute the heat trace, heat content asymptotics, and other spectral functions for manifolds with a high amount of rotational symmetry -- for example on the ball where the metric has complete angular symmetry and only the radial dependence is relevant. The computations then often reduce to 1-dimensional calculations involving Bessel functions. (For other symmetries other special functions are encountered.) In the context of the heat equation, these special case computations can then be fed into known general formulae to compute the undetermined universal coefficients. This method complements the `functorial method' and provides a rapid and efficient means of computation. In quantum field theory the method explained in the book allows for the analysis of Casimir energies, effective actions and partition sums for a large class of examples. NEWLINENEWLINENEWLINEThe book is well written and provides a pleasing introduction to the subject for both the mathematician and the physicist alike.