On the expansion of the resolvent for elliptic boundary contact problems

Details On the expansion of the resolvent for elliptic boundary contact problems On the expansion of the resolvent for elliptic boundary contact problems

2008-01-24

arxiv.org/abs/0801.3852v1

Mathematics

Spectral Theory

Scientific paper

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 $\C$.

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