On the expansion of the resolvent for elliptic boundary contact problems

Abstract

Let A be an elliptic operator on a compact manifold with boundary \({\overline{M}}\) , and let \({\wp:\partial\overline{M} \to Y}\) 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 \({Tr_{L^2} (A_C-\lambda)^{-N}}\) 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 \({\mathbb {C}}\).