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 $L^2(mathbb{R}^2; dt dx)$ of the type $D_A = (d/dt) + A$, where $A = int^{oplus}_{mathbb{R}} dt , A(t)$, and the family of self-adjoint operators $A(t)$ in $L^2(mathbb{R}; dx)$ is explicitly given by $A(t) = - i (d/dx) + theta(t) phi(cdot)$, $t in mathbb{R}$. Here $phi: mathbb{R} to mathbb{R}$ 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 $A_- = - i (d/dx)$, $A_+ = - i (d/dx) + phi(cdot)$ 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 $H_1 = {D_A}^* D_A$, $H_2 = D_A {D_A}^*$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$ W_r(D_A) = lim_{lambda to 0} (- lambda) {rm tr}_{L^2(mathbb{R}^2; dtdx)}((H_1 - lambda I)^{-1} - (H_2 - lambda I)^{-1}). $$ In the concrete example at hand, we prove $$ W_r(D_A) = xi(0_+; H_2, H_1) = xi(0; A_+, A_-) = 1/(2 pi) int_{mathbb{R}} dx , phi(x). $$ Here $xi(, cdot , ; S_2, S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $xi(lambda; H_2, H_1) = 0$, $lambda < 0$.