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

Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [12]. Natural examples arise from (1 + 1)-dimensional differential operators using the model operator DA in L2(R2;dtdx) of the type DA=ddt+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)=-iddx+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-=-iddx,A+=-iddx+φ(·) as t→∓∞, respectively. The interesting feature is that DA violates the relative trace class condition introduced in , 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 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-)=12π∫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.