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.Comment: 26 pages, 1 figure