Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations

Fuda, Toru; Funakawa, Daiju; Suzuki, Akito

doi:10.1063/1.5035300

Cited by 27 publications

(23 citation statements)

References 39 publications

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“…We will prove the existence and the completeness of wave operators for U and a free evolution operator U 0 , we will show the existence of the asymptotic velocity for U, and we will finally establish a weak limit theorem for U. Other interesting related topics such as the existence and the robustness of a bound state localised around the phase boundary or a weak limit theorem for the split-step quantum walk with θ 1 = 0 are considered in [14] and [13], respectively.…”

Section: Introductionmentioning

confidence: 99%

Quantum walks with an anisotropic coin I: spectral theory

2017

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We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we determine the essential spectrum of U, we show the existence of locally U-smooth operators, we prove the discreteness of the eigenvalues of U outside the thresholds, and we prove the absence of singular continuous spectrum for U. Our analysis is based on new commutator methods for unitary operators in a two-Hilbert spaces setting, which are of independent interest.Comment: 26 page

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Section: Introductionmentioning

confidence: 99%

Quantum walks with an anisotropic coin I: spectral theory

2017

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“…For this topic for position-independent cases, see Tate [36]. We also mention Segawa-Suzuki [34], Fuda et al [11] and [12]. The authors used the discriminant operator which is a kind of Hamiltonians generating the time evolution operators of some DTQWs.…”

Section: 2mentioning

confidence: 99%

Generalized eigenfunctions and scattering matrices for position-dependent quantum walks

Morioka

2019

Rev. Math. Phys.

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We study the spectral analysis and the scattering theory for time evolution operators of position-dependent quantum walks. Our main purpose of this paper is construction of generalized eigenfunctions of the time evolution operator. Roughly speaking, the generalized eigenfunctions are not square summable but belong to ℓ ∞ -space on Z. Moreover, we derive a characterization of the set of generalized eigenfunctions in view of the time-harmonic scattering theory. Thus we show that the S-matrix associated with the quantum walk appears in the singularity expansion of generalized eigenfunctions.

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“…This is a phenomenon that the existence probability of a quantum walker is strictly positive after infinitely many time evolutions on some positions. Conditions on the localization are considered in [9,10,11,18] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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

Matsuzawa¹,

Suzuki²,

Tanaka³

et al. 2021

Preprint

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Split-step quantum walks possess chiral symmetry. From a supercharge, that is the imaginary part of the time evolution operator, the Witten index can be naturally defined and it gives the lower bound of the dimension of eigenspaces of 1 or −1. The first three authors studied the Witten index as a Fredholm index under the condition that the supercharge satisfies the Fredholm condition. In this paper, we establish the Witten index formula under the non-Fredholm condition. To derive it, we employ the spectral shift function induced by the fourth-order difference operator with a rank one perturbation on the half-line. Under two-phase limit conditions and trace conditions, the Witten index only depends on parameters of two-side limits. Especially, half-integer indices appear.

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Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations

Cited by 27 publications

References 39 publications

Quantum walks with an anisotropic coin I: spectral theory

Quantum walks with an anisotropic coin I: spectral theory

Generalized eigenfunctions and scattering matrices for position-dependent quantum walks

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

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