We review previous work on spectral flow in connection with certain self-adjoint model operators {A(t)}_t∈R on a Hilbert space H, joining endpoints A_±, and the index of the operator D_A^= (d/d t) + A acting in L^2(R; H), where A denotes the operator of multiplication (A f)(t) = A(t)f(t). In this article we review what is known when these operators have some essential spectrum and describe some new results in terms of associated spectral shift functions. We are especially interested in

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... s to non-Fredholm situations, replacing the Fredholm index by the Witten index, and use a particular (1+1)-dimensional model setup to illustrate our approach based on spectral shift functions.