The Spectral shift function and the Witten index

We survey the notion of the spectral shift function of a pair of self-adjoint operators and recent progress on its connection with the Witten index. We also describe a proof of Krein's Trace Theorem that does not use complex analysis [53] and develop its extension to general $\sigma$-finite von Neumann algebras $\mathcal{M}$ of type II and unbounded perturbations from the predual of $\mathcal{M}$. We also discuss the connection between the theory of the spectral shift function and index theory for certain model operators. We start by introducing various definitions of the Witten index, (an extension of the notion of Fredholm index to non-Fredholm operators). Then we study the model operator $D_{A^{}} = (d/dt) + A$ in $L^2(\mathbb{R};\mathcal{H})$ 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 $f \in L^2(\mathbb{R};\mathcal{H})$. The setup permits the operator family $A(t)$ on $\mathcal{H}$ to be an unbounded relatively trace class perturbation of the unbounded self-adjoint operator $A_-$, and no discrete spectrum assumptions are made on the asymptotes $A_{\pm}$. When $A_{\pm}$ are boundedly invertible, it is shown that $D_{A^{}}$ is Fredholm and its index can be computed as $\xi(0; A_+, A_-)$. When $0\in\sigma(A_+)$ (or $0\in\sigma(A_-)$), the operator $D_{A^{}}$ ceases to be Fredholm. However, if $0$ is a right and a left Lebesgue point of $\xi(\,\cdot\,\, ; A_+, A_-)$, the resolvent regularized Witten index $W_r(D_{A^{}})$ is given by $W_r(D_{A^{}}) = \xi(0_+; |D_{A^{*}}|^2, |D_{A^{}}|^2) = [\xi(0_+; A_+,A_-) + \xi(0_-; A_+, A_-)]/2$. We also study a special example, when the perturbation of the unbounded self-adjoint operator $A_-$ is not assumed to be relatively trace class.