On the Witten index in terms of spectral shift functions

Abstract

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}}) \\ & = ξ_L(0_+; \mathbf{H_2}, \mathbf{H_1}) \\ & = [ξ_L(0_+; A_+,A_-) + ξ_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.

47 pages, Remark 3.6 replaced by Example 3.6

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