ADIABATIC LIMITS OF THE ETA-INVARIANTS THE ODD-DIMENSIONAL ATIYAH-PATODI-SINGER PROBLEM

We study the eta-invariant of boundary value problems of Atiyah-Patodi-Singer type. We prove the formula for the spectral flow of the families over S1. Assuming a product structure in a collar neighbourhood of the boundary, we show that the eta-invariant behaves the same way as on a closed manifold. We also study the "adiabatic" limit of the eta-invariant. In fact, we present a general method for the calculation of the "adiabatic" limits of the spectral invariants. In nice cases we are able to split them into a contribution from the interior, one from the cylinder, and an error term. Then we show that the error term disappears with the increasing length of the cylinder.