Second-order operators with a non-scalar leading symbol (non-Laplace type operators)

In [12,13] (together with T. Branson) we initiate a systematic study of a wide class of natural second-order partial differential operators $F=\nabla^* a\nabla+Q=-a^{\mu\nu}\nabla_\mu\nabla_\nu+Q$ acting on sections of spin-tensor vector bundles over a compact Riemannian manifold without boundary whose leading symbols are not of Laplace type, i.e. we do not assume that $a^{\mu\nu}=g^{\mu\nu}E$ for some automorphism E. In [12] we studied the asymptotic expansions of the corresponding resolvent and the heat kernel. The heat kernel and the Green's function are constructed explicitly in the leading order, and the first two coefficients of the heat kernel asymptotic expansion are computed explicitly. We also construct a new semi-classical ansatz as well as the complete recursion system for the heat kernel of non-Laplace type operators. Some particular cases are studied in more detail.

In [13] we define a discrete leading symbol for such operators which may be computed pointwise, or from spectral asymptotics. We indicate how this can be applied to the computation of another kind of spectral asymptotics, namely asymptotic expansions of fundamental solutions, and to the computation of conformally covariant operators.