The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition

Abstract

Split-step quantum walks possess chiral symmetry. From a supercharge, that is the imaginary part of the time evolution operator, the Witten index can be naturally defined and it gives the lower bound of the dimension of eigenspaces of 1 or −1. The first three authors studied the Witten index as a Fredholm index under the condition that the supercharge satisfies the Fredholm condition. In this paper, we establish the Witten index formula under the non-Fredholm condition. To derive it, we employ the spectral shift function induced by the fourth-order difference operator with a rank one perturbation on the half-line. Under two-phase limit conditions and trace conditions, the Witten index only depends on parameters of two-side limits. Especially, half-integer indices appear.