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.
CITATION STYLE
Matsuzawa, Y., Suzuki, A., Tanaka, Y., Teranishi, N., & Wada, K. (2023). The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition. Reviews in Mathematical Physics, 35(6). https://doi.org/10.1142/S0129055X23500101
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