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

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}$^2; dt dx)$ofthetype$D_A = (d/dt) + A$,where$A = \int^{\oplus}_{$\mathbb{R}$} dt , A(t)$,andthefamilyofself-adjointoperators$A(t)$in$L^2($\mathbb{R}$; dx)$isexplicitlygivenby$A(t) = - i (d/dx) + \theta(t) \phi(\cdot)$,$t \in $\mathbb{R}$$\mathrm{.}Here$\phi: $\mathbb{R}$ \to $\mathbb{R}$$hastobeintegrableon$$\mathbb{R}$$and$\theta: $\mathbb{R}$ \to $\mathbb{R}$$tendstozeroas$t \to - \infty$andto$1$as$t \to + \infty$\mathrm{.}Inparticular,$A(t)$ has asymptotes in the norm resolvent sense $A_{-}=-i(d\mathrm{/}dx)$, $A_{+}=-i(d\mathrm{/}dx)+\varphi (\cdot )$ as $t\to \mp \mathrm{\infty }$. Since $D_A$ violates the relative trace class condition introduced in [9], we now employ a new approach based on an approximation technique. The approximants do fit the framework of [9] and lead to the following results: Introducing $H_1 = {D_A}^* D_A$,$H_2 = D_A {D_A}^*$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$W_r(D_A)=\underset{\lambda \to 0}{\lim }(-\lambda ){\mathrm{t}\mathrm{r}}_{L^2({\mathbb{R}}^2;dtdx)}((H_1-\lambda I{)}^{-1}-(H_2-\lambda I{)}^{-1})\mathrm{.}$$ In the concrete example at hand, we prove $$W_r(D_A)=\xi (0_{+};H_2,H_1)=\xi (0;A_{+},A_{-})=1\mathrm{/}(2\pi ){\int }_{\mathbb{R}}dx\varphi (x)\mathrm{.}$$ Here $\xi (\cdot ;S_2,S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $\xi (\lambda ;H_2,H_1)=0$, $\lambda <0$.