On index theory for non-Fredholm operators: A (1 + 1)-dimensional example

Abstract

Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from (1 + 1)-dimensional differential operators using the model operator DA in L2(R2; dtdx) of the type DA = d dt + A, where A = ´ ⊕ R dtA(t), and the family of self-adjoint operators A(t) in L2(R; dx) studied here is explicitly given by A(t) = −i d dx + θ(t)φ(·), t ∈ R. Here φ : R → R has to be integrable on R and θ : R → R tends to zero as t → −∞ and to 1 as t → +∞ (both functions are subject to additional hypotheses). In particular, A(t), t ∈ R, has asymptotes (in the norm resolvent sense) A− = −i d dx , A+ = −i d dx + φ(·) as t → ∓∞, respectively. The interesting feature is that DA violates the relative trace class condition introduced in [9], Hypothesis 2.1 (iv)]. A new approach adapted to differential operators of this kind is given here using an approximation technique. The approximants do fit the framework of [9] enabling the following results to be obtained. Introducing H1 = D∗ A DA, H2 = DA D∗ A, we recall that the resolvent regularized Witten index of DA, denoted by Wr(DA), is defined by Wr(DA) = lim λ↑0 (−λ)trL2(R2;dtdx) (H1 − λI) −1 − (H2 − λI) −1 whenever this limit exists. In the concrete example at hand, we prove Wr(DA) = ξ (0+; H2, H1) = ξ (0; A+, A−) = 1 2π ˆ R dx φ(x). Here ξ (·; S2, S1) denotes the spectral shift operator for the pair of self-adjoint operators (S2, S1), and we employ the normalization, ξ (λ; H2, H1) = 0, λ < 0.