  Let $\{A(t)\}_{t \in \mathbb{R}}$ be a path of self-adjoint Fredholm
operators in a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$ as $t \to
\pm \infty$. Computing the index of the operator $D_A= (d/d t) + A$ acting in
$L^2(\mathbb{R}; \mathcal{H})$, where $A = \int_{\mathbb{R}}^{\oplus} dt \,
A(t)$, and its relation to spectral flow along this path, has a long history.
While most of the latter focuses on the case where $A(t)$ all have purely
discrete spectrum, we now particularly study situations permitting essential
spectra.
  Introducing $H_1={D_A}^* D_A$ and $H_2=D_A {D_A}^*$, we consider spectral
shift functions $\xi(\, \cdot \,; A_+, A_-)$ and $\xi(\, \cdot \, ; H_2, H_1)$
associated with the pairs $(A_+, A_-)$ and $(H_2,H_1)$. Assuming $A_+$ to be a
relatively trace class perturbation of $A_-$ and $A_{\pm}$ to be Fredholm, the
value $\xi(0; A_-, A_+)$ was shown in [14] to represent the spectral flow along
the path $\{A(t)\}_{t\in \mathbb{R}}$ while that of $\xi(0_+; H_1,H_2)$ yields
the Fredholm index of $D_A$. The fact, proved in [14], that these values of the
two spectral functions are equal, resolves the index = spectral flow question
in this case.
  When the path $\{A(t)\}_{t \in \mathbb{R}}$ consists of differential
operators, the relatively trace class perturbation assumption is violated. The
simplest assumption that applies (to differential operators in
(1+1)-dimensions) is a relatively Hilbert-Schmidt perturbation. This is not
just an incremental improvement. In fact, the approximation method we employ
here to make this extension is of interest in any dimension. Moreover we
consider $A_\pm$ which are not necessarily Fredholm and we establish that the
relationships between the two spectral shift functions for the pairs $(A_+,
A_-)$ and $(H_2,H_1)$ found in all of the previous papers [9], [14], and [22]
can be proved in the non-Fredholm case.
