An APS index theorem for even-dimensional manifolds with non-compact boundary

We study the index of the APS boundary value problem for a strongly Callias-type operator D on a complete Riemannian manifold M . We use this index to define the relative η-invariant η(A₁, A₀) of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the ηinvariants of A₁ and A₀ are not defined, the relative η-invariant behaves as if it were the difference η(A₁) − η(A₀). We also define the spectral flow of a family of such operators and use it to compute the variation of the relative η-invariant.