Abstract

Using the general formalism of [12], a study of index theory for nonFredholm 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=∫⊕RdtA(t), and the family of selfadjoint operators A(t) in L2(R;dx) 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→+∞. In particular, A(t) 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=DA∗DA, H2=DADA∗, we recall that the resolvent regularized Witten index of DA, denoted by Wr(DA), is defined by
Wr(DA)=limλ→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)∫Rdxϕ(x).
Here ξ(⋅;S2,S1), denotes the spectral shift operator for the pair of selfadjoint operators (S2,S1), and we employ the normalization, ξ(λ;H2,H1)=0, λ<0.