  We study the model operator $\mathbf{D}_{\mathbf{A}} = (d/dt) + \mathbf{A}$
in $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path
$\{A(t)\}_{t=-\infty}^{\infty}$, where $(\mathbf{A} f)(t) = A(t) f(t)$ for
a.e.\ $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$
(with $\mathcal{H}$ a separable, complex Hilbert space). Denoting by $A_{\pm}$
the norm resolvent limits of $A(t)$ as $t \to \pm \infty$, our setup permits
$A(t)$ in $\mathcal{H}$ to be an unbounded, relatively trace class perturbation
of the unbounded self-adjoint operator $A_-$, and no discrete spectrum
assumptions are made on $A_{\pm}$.
  We introduce resolvent and semigroup regularized Witten indices of
$\mathbf{D}_{\mathbf{A}}$, denoted by $W_r$ and $W_s$, and prove that these
regularized indices coincide with the Fredholm index of
$\mathbf{D}_{\mathbf{A}}$ whenever the latter is Fredholm.
  In situations where $\mathbf{D}_{\mathbf{A}}$ ceases to be a Fredholm
operator in $L^2(\mathbb{R};\mathcal{H})$ we compute its resolvent (resp.,
semigroup) regularized Witten index in terms of the spectral shift function
$\xi(\,\cdot\,;A_+,A_-)$ associated with the pair $(A_+, A_-)$ as follows:
Assuming $0$ to be a right and a left Lebesgue point of $\xi(\,\cdot\,\, ; A_+,
A_-)$, denoted by $\xi_L(0_+; A_+,A_-)$ and $\xi_L(0_-; A_+, A_-)$, we prove
that $0$ is also a right Lebesgue point of $\xi(\,\cdot\,\, ; \mathbf{H_2},
\mathbf{H_1})$, denoted by $\xi_L(0_+; \mathbf{H_2}, \mathbf{H_1})$, and that
\begin{align*} W_r(\mathbf{D}_{\mathbf{A}}) &= W_s(\mathbf{D}_{\mathbf{A}}) \\
& = \xi_L(0_+; \mathbf{H_2}, \mathbf{H_1}) \\ & = [\xi_L(0_+; A_+,A_-) +
\xi_L(0_-; A_+, A_-)]/2, \end{align*} the principal result of this paper.
  In the special case where $\dim(\mathcal{H}) < \infty$, we prove that the
Witten indices of $\mathbf{D}_{\mathbf{A}}$ are either integer, or
half-integer-valued.
