On the expansion of the resolvent for elliptic boundary contact problems
Abstract
Description
Let $A$ be an elliptic operator on a compact manifold with boundary $M$, and let $\wp : \partial\M \to Y$ be a covering map, where $Y$ is a closed manifold. Let $A_C$ be a realization of $A$ subject to a coupling condition $C$ that is elliptic with parameter in the sector $Λ$. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary.
We prove that the resolvent trace $\Tr_{L^2} (A_C-λ)^{-N}$ for $N$ sufficiently large has a complete asymptotic expansion as $|λ| \to \infty$, $λ\in Λ$. In particular, the heat trace $\Tr_{L^2}e^{-tA_C}$ has a complete asymptotic expansion as $t \to 0^+$, and the $ζ$-function has a meromorphic extension to $\C$.