Trace formulas for a class of non-Fredholm operators: A review

Take a one-parameter family of self-adjoint Fredholm operators {A(t)}t ∈ℝ on a Hilbert space ℋ, joining endpoints A±. There is a long history of work on the question of whether the spectral flow along this path is given by the index of the operator DA = (d/dt) + A acting in L2(∈ℝ; ℋ), where A denotes the multiplication operator (Af)(t) = A(t)f(t) for f dom(A). Most results are about the case where the operators A have compact resolvent. In this article, we review what is known when these operators have some essential spectrum and describe some new results. Using the operators H₁ = D_A^∗D_A, H₂ = DAD_A^∗, an abstract trace formula for Fredholm operators with essential spectrum was proved in [23], extending a result of Pushnitski [35], although, still under strong hypotheses on A: trL2(∈ℝ;ℋ)((H₂ - zI)-1 - (H₁ - zI)-1) = 1 2ztrL2(ℋ)(gz(A+) - gz(A-)), where gz(x) = x(x2 - z)-1/2, x ∈ℝ, z ℂ\[0,∞). Associated to the pairs (H₂,H₁) and (A+,A-) are Krein spectral shift functions (H₂,H₁) and (A+,A-), respectively. From the trace formula, it was shown that there is a second, Pushnitski-type, formula: (γ; H₂,H₁) = 1 π-γ1/2γ1/2 (; A+,A-)d (γ - 2)1/2 for a.e. γ > 0. This can be employed to establish the desired equality, Fredholm index = (0; A+,A-) = spectral flow. This equality was generalized to non-Fredholm operators in [14] in the form Witten index = [R(0; A+,A-) + L(0; A+,A-)]/2, replacing the Fredholm index on the left-hand side by the Witten index of DA and (0; A+,A-) on the right-hand side by an appropriate arithmetic mean (assuming 0 is a right and left Lebesgue point for (A+,A-) denoted by R(0; A+,A-) and L(0; A+,A-), respectively). But this applies only under the restrictive assumption that the endpoint A+ is a relatively trace class perturbation of A- (ruling out general differential operators). In addition to reviewing this previous work, we describe in this article some extensions using a (1 + 1)-dimensional setup, where A± are non-Fredholm differential operators. By a careful analysis we prove, for a class of examples, that the preceding trace formula still holds in this more general situation. Then we prove that the Pushnitski-type formula for spectral shift functions also holds and this then gives the equality of spectral shift functions in the form (γ; H₂,H₁) = (; A+,A-)for a.e. γ > 0 and a.e. ∈ℝ, for the (1 + 1)-dimensional model operator at hand. This shows that neither the relatively trace class perturbation assumption nor the Fredholm assumption are required if one works with spectral shift functions. The results support the view that the spectral shift function should be a replacement for the spectral flow in certain non-Fredholm situations and also point the way to the study of higher-dimensional cases. We discuss the connection with summability questions in Fredholm modules in an appendix.