On the L^pindex of spin Dirac operators on conical manifolds

Abstract

Abstract. We compute the index of the Dirac operator on a spin Riemannian manifold with conical singularities, acting from L p (Σ + ) to L q (Σ − ) with p, q > 1. When 1 + n/p − n/q > 0 we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at (n + 1)/2 − n/q instead of 0 in the definition of the eta invariant. In particular we reprove Chou's formula for the L 2 index. For 1 + n/p − n/q ≤ 0 the index formula contains an extra term related to the Calderón projector.

Exotic Expansions and Pathological Properties of ζ-Functions on Conic Manifolds

“…(The condition A Γ ≥ − 1 4 is necessary otherwise ∆ is not bounded below [5,8].) Laplacians on forms and squares of Dirac operators on conic manifolds [5,10,12,13,36,44,50] are examples of second order regular singular operators. We can also deal with the case when M has boundary components up to which ∆ is smooth; at such components, put local boundary conditions.…”

The ubiquitous ζ-function and some of its ‘usual’ and ‘unusual’ meromorphic properties

“…Other examples include the Laplacian on forms and squares of Dirac operators on conic manifolds [8,12,13,14,36,41,45].…”