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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 [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2(\mathbb{R}^2; dt dx)$ of the type $D_A = (d/dt) + A$, where $A = \int^{\oplus}_{\mathbb{R}} dt \, A(t)$, and the family of self-adjoint operators $A(t)$ in $L^2(\mathbb{R}; dx)$ is explicitly given by $A(t) = - i (d/dx) + θ(t) ϕ(\cdot)$, $t \in \mathbb{R}$. Here $ϕ: \mathbb{R} \to \mathbb{R}$ has to be integrable on $\mathbb{R}$ and $θ: \mathbb{R} \to \mathbb{R}$ tends to zero as $t \to - \infty$ and to $1$ as $t \to + \infty$. In particular, $A(t)$ has asymptotes in the norm resolvent sense $A_- = - i (d/dx)$, $A_+ = - i (d/dx) + ϕ(\cdot)$ as $t \to \mp \infty$. Since $D_A$ violates the relative trace class condition introduced in [9], we now employ a new approach based on an approximation technique. The approximants do fit the framework of [9] and lead to the following results: Introducing $H_1 = {D_A}^* D_A$, $H_2 = D_A {D_A}^*$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$ W_r(D_A) = \lim_{λ\to 0} (- λ) {\rm tr}_{L^2(\mathbb{R}^2; dtdx)}((H_1 - λI)^{-1} - (H_2 - λI)^{-1}). $$ In the concrete example at hand, we prove $$ W_r(D_A) = ξ(0_+; H_2, H_1) = ξ(0; A_+, A_-) = 1/(2 π) \int_{\mathbb{R}} dx \, ϕ(x). $$ Here $ξ(\, \cdot \, ; S_2, S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $ξ(λ; H_2, H_1) = 0$, $λ< 0$.