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arXiv:2007.12832 · Reviews in Mathematical Physics

Absence of singular continuous spectra and embedded eigenvalues for one-dimensional quantum walks with general long-range coins

Masaya Maeda, Akito Suzuki, Kazuyuki Wada

Published Mar 31, 2022• 3 citations
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Abstract

This paper is a continuation of work by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the gauge transformation and construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case.

DOI
10.1142/s0129055x22500167

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This paper is a continuation of work by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the gauge transformation and construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case.
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Mathematical analysis of the spectral properties of the time-evolution operator in quantum walks is essential for understanding key dynamical behaviors such as localization and long-term evolution. The inhomogeneous three-state case, in particular, poses substantial analytical challenges due to its higher internal degrees of freedom and the absence of translational invariance. We develop a general framework for the spectral analysis of three-state quantum walks on the one-dimensional lattice with arbitrary time-evolution operators. Our approach is based on a transfer-matrix formulation that reduces the infinite-dimensional eigenvalue problem to a tractable system of two-dimensional recursions, enabling exact characterization of eigenstates. This framework applies broadly to space-inhomogeneous models, including those with finite defects and two-phase structures. We rigorously derive necessary and sufficient conditions for the existence of point spectrum, along with a complete description of the corresponding eigenvalues and eigenstates, which are known to underlie quantum localization phenomena. Furthermore, we give a complete spectral decomposition—discrete spectrum, flat-band eigenvalues (of infinite multiplicity), and absolutely continuous spectrum—with explicit characterization of each component. Using this method, we perform exact numerical analyses of the Fourier walk with spatial inhomogeneity, revealing the emergence of localization despite its delocalized nature in the homogeneous case. Our results provide mathematical tools and physical insights into the structure of quantum walks, offering a systematic path for identifying and characterizing localized quantum states in complex quantum systems.
We consider quantum walks with position dependent coin on 1D lattice Z\mathbb{Z}Z. The dispersive estimate ∥UtPcu0∥l∞≲(1+∣t∣)−1/3∥u0∥l1\| U^{t} P_{c} u_0\|_{l^{\infty}} \lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}∥UtPc​u0​∥l∞​≲(1+∣t∣)−1/3∥u0​∥l1​ is shown under l1,1l^{1,1}l1,1 perturbation for the generic case and l1,2l^{1,2}l1,2 perturbation for the exceptional case, where UUU is the evolution operator of a quantum walk and PcP_{c}Pc​ is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.
The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.
We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators U(C)U(C)U(C) on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix C∈U(3)C\in U(3)C∈U(3), and study their spectral properties. For circulant unitary coin matrices CCC, we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for U(C)U(C)U(C). This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices CCC which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of U(C)U(C)U(C) is pure point.
We examined three different types of one-dimensional quantum walks, all exhibit some degree of inhomogeneity. The first is inhomogeneous only insofar as the coin operator differs at even and odd positions along the line. The second has a coin operator that is inhomogeneous and oscillatory in time. The third has a coin operator that is inhomogeneous and oscillatory in space, differing at every position along the line. Much progress is made towards attaining asymptotics for the first walk, although our analytical solution does not yet agree with the numerics. An asymptotic approach is attempted for the second walk, although it seems that this approach may not be well- suited to the walk. The third walk, it seems, requires more machinery, although progress is made using symmetry and a pentadiagonal transition operator is obtained. This general pentadiagonal operator gives hope for much progress in the near future.