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 D_A = ∂/∂t + A acting on L²(ℝ;H), where A denotes the multiplication operator (Af)(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. Introducing the operators H₁ = D∗_AD_A and H₂ = D_AD^∗_A, we consider spectral shift functions, denoted by ξ(·;A₊,A₋) and ξ (·;H₂,H₁) associated with the pairs (A₊,A₋) and (H₂,H₁). Under the restrictive hypotheses that A₊ is a relatively trace class perturbation of A₋, a relationship between these spectral shift functions was proved in [14], for certain operators A_± with essential spectrum, extending a result of Pushnitski [22]. Moreover, assuming A_± to be Fredholm, the value ξ (0;A₋,A₊) was shown to represent the spectral flow along the path {A(t)}_t∈ℝwhile that of ξ (0₊;H₁,H₂) yields the Fredholm index of D_A. The fact, proved in [14], that these values of the two spectral functions are equal, resolves the index = spectral flow question in this case. This relationship between spectral shift functions was generalized to non-Fredholm operators in [9] again under the relatively trace class perturbation hypothesis. In this situation it asserts that the Witten index of D_A, denoted by W_r(D_A), a substitute for the Fredholm index in the absence of the Fredholm property of D_A, is given by (Formula Presented). Here one assumes that ξ (·;A₋,A₊) possesses a right and left Lebesgue point at 0 denoted by ξ_L(0_±;A₊,A₋) (and similarly for ξ_L(0₊;H₂,H₁)). When the path {A(t)}_t∈ℝ consists of differential operators, the relatively trace class perturbation assumption is violated. The simplest assumption that applies (to differential operators in 1+1 dimensions) is to admit relatively Hilbert-Schmidt perturbations. This is not just an incremental improvement. In fact, the method we employ here to make this extension is of interest in any dimension. Moreover we consider A_± which are not necessarily Fredholm and we establish that the relationships between the two spectral shift functions found in all of the previous papers [9],[14], and [22], can be proved, even in the non-Fredholm case. The significance of our new methods is that, besides being simpler, they also allow a wide class of examples such as pseudodifferential operators in higher dimensions. Most importantly, we prove the above formula for the Witten index in the most general circumstances to date.