Green functions of higher-order differential operators

Abstract

The Green functions of the partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian manifold are investigated via the heat kernel methods. We study the resolvent of a special class of higher-order operators formed by the products of second-order operators of Laplace type defined with the help of a unique Riemannian metric but with different bundle connections and potential terms. The asymptotic expansion of the Green functions near the diagonal is studied in detail in any dimension. As a by-product a simple criterion for the validity of the Huygens principle is obtained. It is shown that all the singularities as well as the non-analytic regular parts of the Green functions of such high-order operators are expressed in terms of the usual heat kernel coefficients $a_k$ for a special Laplace type second-order operator.

26 pages, LaTeX, 65 KB, no figures, some misprints and small mistakes are fixed, final version to appear in J. Math. Phys. (May, 1998)

Cited by in corpus (13)

• Scattering on compact manifolds with infinitely thin horns
• Gauged Gravity via Spectral Asymptotics of non-Laplace type Operators
• Heat Kernel Asymptotics of Operators with Non-Laplace Principal Part
• Heat kernel for higher-order differential operators and generalized exponential functions
• On-diagonal singularities of the Green functions for Schroedinger operators
• Vortices and a TQFT for Lefschetz fibrations on 4-manifolds
• On the photon Green functions in curved space-time
• Higher Derivative Fermionic Field Theories
• Fourth-Order Operators on Manifolds with Boundary
• Quantum heat traces
• Mathematical Tools for Calculation of the Effective Action in Quantum Gravity
• Universal Self-Similar Attractor in the Bending-Driven Leveling of Thin Viscous Films
• Resolvent expansions on hybrid manifolds