An index theorem for split-step quantum walks

Keisuke Asahara, Daiju Funakawa, Motoki Seki +1 more

Akihiro Narimatsu, Hiromichi Ohno, Kazuyuki Wada

Toru Fuda, Akihiro Narimatsu, Kei Saito +1 more

Toru Fuda, Daiju Funakawa, Satoshi Sasayama +1 more

Yasumichi Matsuzawa, Akito Suzuki, Yohei Tanaka +2 more

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 [Formula: see text] and [Formula: see text] 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.

Toru Fuda, Daiju Funakawa, Akito Suzuki

This paper proves a weak limit theorem for a one-dimensional split-step quantum walk and investigates the limit density function. In the density function, the difference between two Konno's functions appears.