  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 $A(t) = - i (d/dx) + \theta(t) \phi(\cdot)$, $t \in
\mathbb{R}$. Here $\phi: \mathbb{R} \to \mathbb{R}$ has to be integrable on
$\mathbb{R}$ and $\theta: \mathbb{R} \to \mathbb{R}$ 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 $A_- = - i (d/dx)$, $A_+ = - i (d/dx) + \phi(\cdot)$
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 $$ W_r(D_A) =
\lim_{\lambda \to 0} (- \lambda) {\rm tr}_{L^2(\mathbb{R}^2; dtdx)}((H_1 -
\lambda I)^{-1} - (H_2 - \lambda I)^{-1}). $$ In the concrete example at hand,
we prove $$ W_r(D_A) = \xi(0_+; H_2, H_1) = \xi(0; A_+, A_-) = 1/(2 \pi)
\int_{\mathbb{R}} dx \, \phi(x). $$ Here $\xi(\, \cdot \, ; S_2, S_1)$, denotes
the spectral shift operator for the pair $(S_2,S_1)$, and we employ the
normalization, $\xi(\lambda; H_2, H_1) = 0$, $\lambda < 0$.
