Spectral Functions

Abstract

This Chapter contains definitions of main spectral functions, lists their properties, and methods of computation. The material includes the Riemann zeta-function, the zeta-function of selfadjoint elliptic second order differential operators (both positive and semidefinite), relations between the zeta-function residues and the heat kernel coefficients. Among less standard issues, a brief description of the spectral density of operators, relation between its large eigenvalue behavior and heat kernel asymptotic expansions is included to fill in existing gaps in textbooks. Determinants of differential operators are introduced by using the Ray-Singer formula followed by other regularization schemes. Then the zeta-function and the determinant of a Dirac operator are introduced. Much space is devoted to variations of determinants generated by transformations of the operators and to spectral functions associated to transformations of so called chiral operators. This material later serves as a basis for calculations of quantum anomalies. The index theorem, which is elaborated in many monographs, here is explained rather briefly to express a merit of the Atiyah-Singer theory and set some definitions.