On the index of differential operators on manifolds with conical singularities
The class of general elliptic pseudodifferential operators on an arbitrary smooth manifold with singularities of conical type subject to some additional conditions of symmetry of the conormal symbol is considered. For operators from this class the index formula as the sum of multiplicities of some spectral points of the conormal symbol and the integral from the Atiyah-Singer form over the smooth part of the manifold is obtained. The main idea of this paper is to construct the regularizer for the considered operator with the help of the Green operator on the singular part of the manifold and the standard regularizer on the smooth part of the underlying manifold. The index formula obtained in the paper for conical singularities is also valid for smooth manifolds with cusp-type singularities. The proof of the index theorem is quite elementary and does not use any complicated algebraic-topological constructions.
scientific article; zbMATH DE number 1157099