Quantum walks with an anisotropic coin I: spectral theory

Abstract

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.

26 pages

References in corpus (6)

• Exploring Topological Phases With Quantum Walks
• Quantum random walks in one dimension
• Quantum walks with an anisotropic coin II: scattering theory
• Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations
• Weak limit theorem for a one-dimensional split-step quantum walk
• Entanglement Properties of Localized States in 1D Topological Quantum Walks

Cited by in corpus (24)

• Quantum walks with an anisotropic coin II: scattering theory
• Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations
• Generalized eigenfunctions and scattering matrices for position-dependent quantum walks
• Continuous limits of linear and nonlinear quantum walks
• The Witten Index for 1D Supersymmetric Quantum Walks with Anisotropic Coins
• Detection of edge defects by embedded eigenvalues of quantum walks
• A Constructive Approach to Topological Invariants for One-dimensional Strictly Local Operators
• Eigenvalues of two-phase quantum walks with one defect in one dimension
• Generalized eigenfunctions for quantum walks via path counting approach
• Absence of wave operators for one-dimensional quantum walks
• Quantum time delay for unitary operators: general theory
• Index Theory of Chiral Unitaries and Split-Step Quantum Walks
• Absence of Bound States for Quantum Walks and CMV Matrices via Reflections
• Asymptotic properties of generalized eigenfunctions for multi-dimensional quantum walks
• Spectral and scattering properties of quantum walks on homogenous trees of odd degree
• An eigenfunction expansion formula for one-dimensional two-state quantum walks
• The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition
• Dynamical Localization and Transport properties of Quantum Walks on the hexagonal lattice
• Two-dimensional quantum central limit theorem by quantum walks
• Index Theorems for One-dimensional Chirally Symmetric Quantum Walks with Asymptotically Periodic Parameters
• The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution
• Spectral analysis of three-state quantum walks with general coin matrices
• Complex translation methods and its application to resonances for quantum walks
• A weak limit theorem for a class of long range type quantum walks in 1d