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

It is recently shown that a split-step quantum walk possesses a chiral symmetry, and that a certain well-defined index can be naturally assigned to it. The index is a well-defined Fredholm index if and only if the associated unitary time-evolution operator has spectral gaps at both $+1$ and $-1.$ In this paper we extend the existing index formula for the Fredholm case to encompass the non-Fredholm case (i.e., gapless case). We make use of a natural extension of the Fredholm index to the non-Fredholm case, known as the Witten index. The aim of this paper is to fully classify the Witten index of the split-step quantum walk by employing the spectral shift function for a rank one perturbation of a fourth order difference operator. It is also shown in this paper that the Witten index can take half-integer values in the non-Fredholm case.