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arXiv · 2111.04108

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

Yasumichi Matsuzawa·Akito Suzuki·Yohei Tanaka·Noriaki Teranishi·Kazuyuki Wada
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Abstract

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.

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Yasumichi Matsuzawa, Akito Suzuki, Yohei Tanaka, Noriaki Teranishi, Kazuyuki Wada. 2022-09-08. The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition. https://doi.org/10.1142/s0129055x23500101

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