On the index of a non-Fredholm model operator

Let {A(t)}t∈R be a path of self-adjoint Fredholm operators in a Hilbert space H , joining endpoints A± as t → ±∞. Computing the index of the operator DA = ∂ /∂ t + A acting on L2(R;H ), where A denotes the multiplication operator (A f)(t) = A(t) f (t) for f ∈ L2(R;H ) , and its relation to spectral flow along this path, has a long history, but it is mostly focussed on the case where the operators A(t) all have purely discrete spectrum.