1. Introduction
The coefficients in the asymptotic expansion of the diagonal heat kernel elements are of fundamental importance in theoretical physics, analysis, spectral geometry and topology of manifolds. They determine one-loop divergences of the effective action, the axial and trace anomalies (DeWitt, 1965; Birrell end Davies, 1982; Brown, 1977; Dowker, 1978; Christensen and Duff, 1979) in quantum field theory, high-temperature expansion of the partition function in statistical physics, the indices of elliptic operators, geometric and topological properties of manifolds (Atyah, Bott and Patodi, 1973; Gilkey, 1974; Minakshisundaram and Pleijel, 1949; Kac, 1966; McKean and Singer, 1967).
The asymptotic expansion we aie interested in, reads (Atyah, Bott and Patodi, 1973; Gilkey, 1974; Seeley, 1967; Greiner, 1971; Gilkey, 1975)
(x\e-tA\x) ~ £ Em(x|A)t^, t -» 0+ , (1.1)
m>0
where Л is a positive elliptic differential operator of the order 2r acting, in general, on the bundle of i-tensors with the values in a vector space V, the base M being the compact closed n-dimensional manifold. The expansion coefficients Em(x\A) are endomorphisms of the fibre at * and prove to be the local covariant quantities of given dimensionality constructed from the coefficients of the operator A, curvatures, torsion and their derivatives. The £m's are called DeWitt-Seeley-Gilkey (DWSG) coefficients though many people contributed into their investigation beginning from Hadamard (see Schimming (1993) for historical background of the question).
The typical examples of operators A used in physics and mathematics are
-П + Х,
(1.2)
