  In this paper we establish a formula, expressing the generalized
Atiyah-Patodi-Singer index in terms of eta invariants of domain-wall massive
Dirac operators, without assuming that the Dirac operator on the boundary is
invertible. Compared with the original Atiyah-Patodi-Singer index theorem, this
formula has the advantage that no global spectral projection boundary
conditions appear. Our main tool is an asymptotic gluing formula for eta
invariants proved by using a splitting principle developed by Douglas and
Wojciechowski in adiabatic limit. The eta invariant splits into a contribution
from the interior, one from the boundary, and an error term vanishing in the
adiabatic limit process.
