On a spectral flow formula for the homological index

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Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators A(t) parametrized by the real line. Then under certain conditions RS95 that include the assumption that the operators D(t)= D+A(t) all have discrete spectrum then the spectral flow along the path D(t) can be shown to be equal to the index of d/dt+D(t) when the latter is an unbounded Fredholm operator on L²(R, H). In GLMST11 an investigation of the index=spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general. In CGPST14a the question of what happens when the Fredholm condition is dropped altogether was investigated. In these circumstances the Fredholm index is replaced by the Witten index. In this paper we take the investigation begun in CGPST14a much further. We show how to generalise a formula known from the setting of the L² index theorem to the non-Fredholm setting. Our main theorem gives a trace formula relating the homological index of CaKa:TIH to an integral formula that is known, for a path of selfadjoint Fredholms with compact resolvent and with unitarily equivalent endpoints, to compute spectral flow. Our formula however, applies to paths of selfadjoint non-Fredholm operators. We interpret this as indicating there is a generalisation of spectral flow to the non-Fredholm setting.