Higher-order differential operators

In [14,15] I investigated the diagonal singularities of the resolvent of differential operators of higher orders. A special class of such operators given by the products $F_{(1)}\cdots F_{(N)}$ of Laplace type operators $F_{(j)}=-\nabla^{*}_{(j)}\nabla_{(j)}+Q_{(j)}$acting on smooth sections of a vector bundle V over a Riemannian manifold was considered. The leading symbol of all operators F_(j) determined by the Riemannian metric of the manifold M is assumed to be the same, whereas the connections $\nabla_{(j)}$ and the endomorphisms Q_(j) are different, so that the operators F_(j) do not commute with each other. The asymptotic expansion of the resolvent near the diagonal is studied in detail. Some new simple explicit formulas were obtained, which express the singularities of the resolvent in terms of the usual well known HMDS-coefficients a_k for a Laplace type operator on a bigger vector bundle (i.e. with an additional matrix structure). As a by-product a simple criterion for the validity of the Huygens principle is obtained.