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



Publication Details

Carey, A. L., Gesztesy, F., Levitina, G., Potapov, D., Sukochev, F. & Zanin, D. (2016). On index theory for non-Fredholm operators: A (1 + 1)-dimensional example. Mathematische Nachrichten, 289 (5-6), 575-609.


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=∫⊕RdtA(t), and the family of self-adjoint 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 self-adjoint operators (S2,S1), and we employ the normalization, ξ(λ;H2,H1)=0, λ<0.

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