Atiyah-Patodi-Singer boundary condition and a splitting formula of a   spectral flow
  We describe a relation between Atiyah-Patodi-Singer boundary condition and a global elliptic boundary condition which naturally appears in formulating a splitting formula for a spectral flow, when we decompose the manifold into two components. Then we give a variant of the splitting formula with the Hoermander index as a correction term. 
Kenro Furutani