On the expansion of the resolvent for elliptic boundary contact problems

TL;DR

This paper establishes a complete asymptotic expansion for the resolvent trace of elliptic boundary contact problems with nonlocal boundary conditions, leading to insights into heat trace asymptotics and zeta function meromorphic extension.

Contribution

It proves the asymptotic expansion of the resolvent trace for elliptic operators with nonlocal boundary conditions respecting a covering structure.

Findings

Resolvent trace admits a complete asymptotic expansion as spectral parameter grows.

Heat trace has a full asymptotic expansion as time approaches zero.

Zeta function extends meromorphically to the complex plane.

Abstract

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 $\C$.