On Index Theory for Non-Fredholm Operators: A $(1+1)$-Dimensional Example

Abstract

Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2({\mathbb{R}}^2;dtdx)$ of the type $D_A=(d/dt)+A$, where $A={\int }_{\mathbb{R}}^{\oplus }dtA(t)$, and the family of self-adjoint operators $A(t)$ in $L^2(\mathbb{R};dx)$ is explicitly given by $A(t)=-i(d/dx)+\theta (t)\varphi (\cdot )$, $t\in \mathbb{R}$. Here $\varphi :\mathbb{R}\to \mathbb{R}$ has to be integrable on $\mathbb{R}$ and $\theta :\mathbb{R}\to \mathbb{R}$ tends to zero as $t\to -\infty$ and to $1$ as $t\to +\infty$. In particular, $A(t)$ has asymptotes in the norm resolvent sense $A_{-}=-i(d/dx)$, $A_{+}=-i(d/dx)+\varphi (\cdot )$ as $t\to \mp \infty$. Since $D_A$ violates the relative trace class condition introduced in [9], we now employ a new approach based on an approximation technique. The approximants do fit the framework of [9] and lead to the following results: Introducing $H_1={D_A}^{\ast }{D}_A$, $H_2=D_A{D_A}^{\ast }$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$W_r(D_A)=\underset{\lambda \to 0}{\lim }(-\lambda ){\mathrm{tr}}_{L^2({\mathbb{R}}^2;dtdx)}((H_1-\lambda I{)}^{-1}-(H_2-\lambda I{)}^{-1}).$$ In the concrete example at hand, we prove $$W_r(D_A)=\xi (0_{+};H_2,H_1)=\xi (0;A_{+},A_{-})=1/(2\pi ){\int }_{\mathbb{R}}dx\varphi (x).$$ Here $\xi (\cdot ;S_2,S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $\xi (\lambda ;H_2,H_1)=0$, $\lambda <0$.

37 pages