Witten Index and spectral shift function

Let $D$ be a selfadjoint unbounded operator on a Hilbert space and let $\{B(t)\}$ be a one parameter norm continuous family of self-adjoint bounded operators that converges in norm to asymptotes $B_\pm$. Then setting $A(t)=D+B(t)$ one can consider the operator $\mathbf{D_A}^{}=d/dt+A(t)$ on the Hilbert space $L_2(\mathbb{R},H)$. We present a connection between the theory of spectral shift function for the pair of the asymptotes $(A_+,A_-)$ and index theory for the operator $\mathbf{D_A}^{}$. Under the condition that the operator $B_+$ is a $p$-relative trace-class perturbation of $A_-$ and some additional smoothness assumption we prove a heat kernel formula for all $t\u3e0$, $$\mathrm{tr}\Big(e^{-t\mathbf{D_A}^{}\mathbf{D_A}^{*}}-e^{-t\mathbf{D_A}^{*}\mathbf{D_A}^{}}\Big)=-\Big(\frac{t}{\pi}\Big)^{1/2}\int_0^1\mathrm{tr}\Big(e^{-tA_s^2}(A_+-A_-)\Big)ds,$$ where $A_s, s\in[0,1]$ is a straight path joining $A_-$ and $A_+$. Using this heat kernel formula we obtain the description of the Witten index of the operator $\mathbf{D_A}^{}$ in terms of the spectral shift function for the pair $(A_+,A_-)$. {\bf Theorem.} \textit{If\, $0$ is a right and a left Lebesgue point of the spectral shift function $\xi(\cdot;A_+,A_-)$ for the pair $(A_+,A_-)$ (denoted by $\xi_L(0_+; A_+,A_-)$ and $\xi_L(0_-; A_+, A_-)$, respectively), then the Witten index $W_s(\mathbf{D_A})$ of the operator $\mathbf{D_A}$ exists and equals $$W_s(\mathbf{D_A})=\frac12\big(\xi(0+;A_+,A_-)+\xi(0-;A_+,A_-)\big).$$} We note that our assumptions include the cases studied earlier. In particular, we impose no assumption on the spectra of $A_\pm$ and we can treat differential operators in any dimension. As a corollary of this theorem we have the following result. {\bf Corollary.} \textit{Assume that the asymptotes $A_\pm$ are boundedly invertible. Then the operator $\mathbf{D_A}$ is Fredholm and for the Fredholm index $\mathrm{index}(\mathbf{D_A})$ of the operator $\mathbf{D_A}$ we have $$\mathrm{index}(\mathbf{D_A})=\xi(0;A_+,A_-)=\mathrm{sf}\{A(t)\}_{t=-\infty}^{+\infty},$$ where $\mathrm{sf}\{A(t)\}_{t=-\infty}^{+\infty}$ denotes the spectral flow along the path $\{A(t)\}_{t=-\infty}^{+\infty}.$

On a conjectured property of the Witten index and an application to Levinson's theorem

A few years ago Fritz Gesztesy raised the issue of whether there was a composition rule for the Witten index analogous to that satisfied by Fredholm operators. In this note we prove a result in this direction and provide an application to Levinson's theorem

On the Global Limiting Absorption Principle for Massless Dirac Operators

We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators $H_0 = \alpha \cdot (-i \nabla)$ for all space dimensions $n \in \mathbb{N}$, $n \geq 2$. This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene. We also prove an essential self-adjointness result for first-order matrix-valued differential operators with Lipschitz coefficients.Comment: 22 page

Trace Formulas for a Class of non-Fredholm Operators: A Review

We review previous work on spectral flow in connection with certain self-adjoint model operators $\{A(t)\}_{t\in \mathbb{R}}$ on a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$, and the index of the operator $D_{A}^{}= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A$ denotes the operator of multiplication $(A f)(t) = A(t)f(t)$. In this article we review what is known when these operators have some essential spectrum and describe some new results in terms of associated spectral shift functions. We are especially interested in extensions to non-Fredholm situations, replacing the Fredholm index by the Witten index, and use a particular $(1+1)$-dimensional model setup to illustrate our approach based on spectral shift functions.Comment: 46 pages. This is a review that in part extends the earlier arXiv:1509.01580, arXiv:1509.01356, and arXiv:1505.04895 submission

Examples

In this chapter we supplement the abstract discussion by several examples for which our general assumption holds and hence the results of Chap. 6

The Spectral Shift Function

As we noted in the introduction the underlying philosophy is that to go beyond the Fredholm case we need a new perspective on index theory. As the spectral shift function calculates spectral flow in the Fredholm case, we regard it as the natural generalisation of spectral flow to the non-Fredholm case because it remains defined without Fredholm assumptions. In this chapter we introduce the spectral shift function and some of its history