Heat Kernel Asymptotics of Operators with Non-Laplace Principal Part

TL;DR

This paper develops a new approach to analyze the heat kernel asymptotics of second-order elliptic operators with non-Laplace principal parts on compact manifolds, providing explicit coefficients and a semi-classical ansatz.

Contribution

It introduces a novel semi-classical ansatz and recursion system for heat kernels of non-Laplace type operators, extending classical methods to more general elliptic operators.

Findings

Explicit leading order heat kernel and Green's function construction

First two heat kernel asymptotic coefficients computed

New recursion system for non-Laplace operators

Abstract

We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part $-\N^\mu\N_\mu$. Our objective is to obtain information on 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. The first two coefficients of the heat kernel asymptotic expansion are computed explicitly. A new semi-classical ansatz as well as the complete recursion system for the heat kernel of non-Laplace type operators is constructed. Some particular cases are studied in more detail.