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The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition.
- Source :
-
Reviews in Mathematical Physics . Jul2023, Vol. 35 Issue 6, p1-41. 41p. - Publication Year :
- 2023
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0129055X
- Volume :
- 35
- Issue :
- 6
- Database :
- Academic Search Index
- Journal :
- Reviews in Mathematical Physics
- Publication Type :
- Academic Journal
- Accession number :
- 164820622
- Full Text :
- https://doi.org/10.1142/S0129055X23500101