  We study the index of the APS boundary value problem for a strongly
Callias-type operator D on a complete Riemannian manifold $M$. We show that
this index is equal to an index on a simpler manifold whose boundary is a
disjoint union of two complete manifolds $N_0$ and $N_1$. If the dimension of
$M$ is odd we show that the latter index depends only on the restrictions $A_0$
and $A_1$ of $D$ to $N_0$ and $N_1$ and thus is an invariant of the boundary.
We use this invariant to define the relative eta-invariant $\eta(A_1,A_0)$. We
show that even though in our situation the eta-invariants of $A_1$ and $A_0$
are not defined, the relative eta-invariant behaves as if it was the difference
$\eta(A_1)-\eta(A_0)$.
