Complex powers on noncompact manifolds and manifolds with singularities
For an elliptic pseudodifferential operator (pdo) A of positive order a family of complex powers \(\{A^ s:\) \(s\in {\mathbb{C}}\}\) is constructed. The kernel function \(k_ s(x,y)\) of \(A^ s\) is a continuous function of (x,y) depending analytically on s for s in a half-plane of \({\mathbb{C}}\). Moreover, in the most interesting case \(x=y\), \(s\mapsto k_ s(x,x)\) is shown to have an analytic extension to a larger half-plane, the domain and the type of possible singularities depending on the asymptotic expansion of the symbol of A. In particular, \(s\mapsto k_ s(x,x)\) can be continued holomorphically into the origin \(s=0\) under corresponding assumptions. Similar results hold for the zeta and eta function of A. The proof uses the calculus of pdo's on SG-compatible manifolds (including manifolds with singularities) based on H. O. Cordes' idea of globally defined symbols and the classical techniques in complex powers of R. Seeley and H. Kumnao-go.
scientific article; zbMATH DE number 4019636
scientific article