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