The index of Callias-type operators with Atiyah-Patodi-Singer boundary   conditions
  We compute the index of a Callias-type operator with APS boundary condition on a manifold with compact boundary in terms of combination of indexes of induced operators on a compact hypersurface. Our result generalizes the classical Callias-type index theorem to manifolds with compact boundary. 
Pengshuai Shi
math.DG