Green functions of higher-order differential operators

Details Green functions of higher-order differential operators Green functions of higher-order differential operators

1997-07-03

arxiv.org/abs/hep-th/9707040v2

J.Math.Phys. 39 (1998) 2889-2909

Physics

High Energy Physics

High Energy Physics - Theory

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

Scientific paper

10.1063/1.532436

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.

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