Localization for magnetic quantum walks

Localization of a multi-dimensional quantum walk with one defect

In this paper, we introduce a multidimensional generalization of Kitagawa's split-step discrete-time quantum walk, study the spectrum of its evolution operator for the case of one defect coins, and prove localization of the walk. Using a spectral mapping theorem, we can reduce the spectral analysis of the evolution operator to that of a discrete Schrödinger operator with variable coefficients, which is analyzed using the Feshbach map.

Anderson Localization for Quasi-Periodic CMV Matrices and Quantum Walks

We consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.

Anderson localization for electric quantum walks and skew-shift CMV matrices

We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the L…

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

For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in \cite{HiSeSu}. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there is no eigenvalues around $\pm 1$ in terms of a discriminant operator. We also provide a criterion for when eigenvalues $\pm 1$ exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces a…

多频拟周期CMV矩阵与量子游走的Anderson局域化

Anderson localization for the multi-frequency quasi-periodic CMV matrices and quantum walks

本文通过构造半代数集并结合雪崩原理与大偏差定理，证明了多频拟周期CMV矩阵的Anderson局域化，并经由酉等价性推广至量子游走。

具有强位移的二维分裂步量子游走的特征值与阈值共振

Eigenvalues and threshold rezonances of a two-dimensional split-step quantum walk with strong shift

本文推导了具有强位移的二维分裂步量子游走的局域化充分条件，通过显式分析函数$f$的零点来表示演化算子的特征值与特征向量，并阐明了特殊情况下特征值渐近趋近于阈值共振的现象。