The resolvent parametrix of the general elliptic linear differential operator: a closed form for the intrinsic symbol

Abstract

Nonrecursive, explicit expressions are obtained for the term of arbitrary order in the asymptotic expansion of the intrinsic symbol of a resolvent parametrix of an elliptic linear differential operator, of arbitrary order and algebraic structure, which acts on sections of a vector bundle over a manifold. Results for the conventional symbol are included as a special case.

Spectral Functions in Mathematics and Physics

“…This is motivated because it is here that the special case of the ball gives rich information. But it is also justified because the calculation of the volume part is nowadays nearly automatic [91,92,93] and these terms do not depend on the boundary conditions [94] and are thus known already for all problems to be dealt with.…”

Derivative expansion for the one-loop effective Lagrangian in QED

“…In Euclidean space this operator is closely connected with the heat kernel operator. Because the matrix elements of U(τ ) cannot be calculated for arbitrary external fields, several approximate regular schemes have been developed: the technique of asymptotic expansion at small values of the proper time (τ → 0) [9][10][11][12][13], covariant perturbative theory [14,15] and, at last, pseudodifferential operator technique [16][17][18][19]. The asymptotic expansion at τ → 0 is equivalent to the asymptotic expansion of the effective action in inverse powers of mass parameter m 2 .…”

Heat kernel coefficients of the Laplace operator on the D-dimensional ball

“…If the manifold M has a boundary ∂M, the coefficients B n in the short-time expansion have both a volume and a boundary part [7,8]. It is usual to write this expansion in the form For the volume part very effective systematic schemes have been developed (see for example [9,10,11]). The calculation of c n , however, is in general more difficult.…”