On the expansion of the resolvent for elliptic boundary contact problems

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 $\Lambda$. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace $\Tr_{L^2} (A_C-\lambda)^{-N}$ for $N$ sufficiently large has a complete asymptotic expansion as $|\lambda| \to \infty$, $\lambda \in \Lambda$. In particular, the heat trace $\Tr_{L^2}e^{-tA_C}$ has a complete asymptotic expansion as $t \to 0^+$, and the $\zeta$-function has a meromorphic extension to $\C$.