On the resolvent of differential operators on conic manifolds
The aim of the paper is to describe the structure of resolvent of a cone differential operator acting between weighted Sobolev spaces on a compact manifold with boundary \(X\). It is proved that if \(A\) is a cone differential operator of order \(m\), fully elliptic with respect to \(\alpha\in {\mathbb R}\) on a close sector \(\Lambda\), then \((A-\lambda)^{-1}\) is \(b\)-pseudodifferential, \(\lambda\mapsto (A-\lambda)^{-1}\) is meromorphic having only finite rank singularities. Moreover a precise description of the asymptotic are given as the spectral parameter \(\lambda\) tends to zero. It is shown how the singularities of the Schwartz kernel of \((A-\lambda)^{-1}\) accumulate near diagonal as \(\lambda\rightarrow \infty\). A trace formula for the operator \(B(A-\lambda)^{-N}\) is given, where \(B\) is a cone differential operator.
scientific article; zbMATH DE number 1925892
scientific article