The index of elliptic operators on manifolds with conical points
The authors offer a new perspective on the index theory of elliptic pseudodifferential operators on manifolds with isolated conical singularities. Their approach is based on the algebra of formal symbols, an example of deformation quantization whose role in index theory is due to \textit{B. V. Fedosov} [Trans. Moscow Math. Soc. 30, 159-240 (1976; Zbl 0349.58006); translation from Trudy Moskov. Mat. Obsc. 30, 159-241 (1974)]. In a departure from previous work on cone Mellin pseudodifferential operators, the authors choose variables in which a deleted neighborhood of the conical point is represented by an infinite cylindrical end. The index formula proven in this paper is given an attractive interpretation in the two-dimensional case. There the formula depends entirely on the principal interior symbol and on the conormal symbol. The formula decomposes as the sum of an Atiyah-Singer integral, an eta-invariant term, and a third term. Calculations with the Cauchy-Riemann operator show that all three terms can be nonzero. The paper's final section interprets the index formula in the context of a symmetry condition on the conormal symbol in arbitrary dimension.
scientific article; zbMATH DE number 1422753