On the expansion of the resolvent for elliptic boundary contact problems (Q2391009)

The very interesting paper under review deals with the expansion of the resolvent for elliptic boundary contact problems. More precisely, let \(A\) be an elliptic operator over a compact manifold with boundary \(\overline{M},\) and let \(\wp:\partial\overline{M} \to Y\) be a covering map with \(Y\) being a closed manifold. Let \(A_{C}\) be a realization of \(A\) subject to a coupling condition \(C\) which is elliptic with parameter in the sector \(\Lambda.\) (Here ``coupling condition'' means a nonlocal boundary condition which respects the covering structure of the boundary.) The author proves that the resolvent trace \(\text{Tr}_{L^2} (A_C-\lambda)^{-N}\) for \(N\) sufficiently large has a complete asymptotic expansion as \(|\lambda|\to \infty,\) \(\lambda \in \Lambda.\) In particular, the heat trace \(\text{Tr}_{L^2}e^{-tA_C}\) has a complete asymptotic expansion as \(t \to 0^+\) and the \(\zeta\)-function has a meromorphic extension to \(\mathbb C.\)