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We compute the Fredholm index, ${\rm ind}(D_A)$, of the operator $D_A =(d/dt) + A$ on $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path$\{A(t)\}_{t=-\infty}^{\infty}$, where $(A f)(t) = A(t) f(t)$ for a.e.$t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$, via thespectral shift function $\xi(\, \cdot \,;A_+,A_-)$ associated with the pair$(A_+, A_-)$ of asymptotic operators $A_{\pm}=A(\pm\infty)$ on the separablecomplex Hilbert space $\mathcal{H}$ in the case when $A(t)$ is generally anunbounded (relatively trace class) perturbation of the unbounded self-adjointoperator $A_-$. We derive a formula (an extension of a formula due toPushnitski) relating the spectral shift function $\xi(\, \cdot \,;A_+,A_-)$ forthe pair $(A_+, A_-)$, and the corresponding spectral shift function $\xi(\,\cdot \,;H_2,H_1)$ for the pair of operators $(H_2,H_1)=(D_A {D_A}^*, {D_A}^*D_A)$ in this relative trace class context. This formula is then used toidentify the Fredholm index of $D_A$ with $\xi(0;A_+,A_-)$. In addition, weprove that ${\rm ind}(D_A)$ coincides with the spectral flowSpFlow$(\{A(t)\}_{t=-\infty}^\infty)$ of the family $\{A(t)\}_{t\in\mathbb{R}}$and also relate it to the (Fredholm) perturbation determinant for the pair$(A_+, A_-)$. We also provide some applications in the context ofsupersymmetric quantum mechanics to zeta function and heat kernel regularizedspectral asymmetries and the eta-invariant.

The index formula and the spectral shift function for relatively trace class perturbations