On the expansion of the resolvent for elliptic boundary contact problems

Let A be an elliptic operator on a compact manifold with boundary M̄ , and let ℘ overline ∂M 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 _l2(A_C-λ^-N for N sufficiently large has a complete asymptotic expansion as λ|→ ∞, λ ∈ δ In particular, the heat trace Tr_L2}e-tA_C has a complete asymptotic expansion as t → 0⁺ , and the ζ -function has a meromorphic extension to C .