Kazuyuki Wada
We show that there exist pairs of two time evolution operators which do not have wave operators in a context of one-dimensional discrete time quantum walks. As a consequence, the borderline between short range type and long range type is decided.
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Absence of Ballistic Transport in Quantum Walks with Asymptotically Reflecting Sites
研究提出零速度的通用条件，证明量子行走中无 ballistic 运输，基于站点序列、间隙结构和局部硬币参数，讨论了确定性和随机情况下的应用。
This paper is a continuation of the paper \cite{W} 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 construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case in \cite{MSSSSdis}.
Absence of Bound States for Quantum Walks and CMV Matrices via Reflections
本文利用反射对称性建立判据，证明分裂步量子行走与CMV矩阵无点谱，并应用于酉几乎Mathieu算子的算术离域化及Thue–Morse子移位生成CMV矩阵的奇异连续谱。
The subject of this paper is a kind of dynamical systems called quantum walks. We study one-dimensional homogeneous analytic quantum walks U. We explain how to identify the space of all the uniform intertwining operators between these walks. We can also determine whether U can be realized by a (not necessarily homogeneous) continuous-time uniform quantum walk on Z. Several examples of quantum walks, which can not be realized by continuous-time uniform quantum walks, are presented. The 4-state Grover walk is one of them. Before stating the main theorems, we clarify the definition of one-dimensi…
In this article we introduce a new type of nonlocal operators and study the Cauchy problem for certain parabolic-type pseudodifferential equations naturally associated to these operators. Some of these equations are the p-adic master equations of certain models of complex systems introduced by Avetisov et al. The fundamental solutions of these parabolic-type equations are transition functions of random walks on the n-dimensional vector space over the field of p-adic numbers. We study some properties of these random walks, including the first passage time.
We derive the weak limit theorem for a class of long range type quantum walks. To do it, we analyze spectral properties of a time evolution operator and prove that modified wave operators exist and are complete.
A quantum walk with a delocalized initial state: contribution from a coin-flip operator
离散时间量子行走的单位演化步长由抛硬币算子和位置移动算子共同决定。量子步行者在许多步之后的行为微妙地取决于硬币翻转算子和步行的初始条件。为了得到这一行为，我们得到了许多从定域初态开始的量子行走的长时间极限分布。在本文中，我们计算了具有非定域初态而非定域初态的两态量子行走的极限分布，并讨论了行走如何依赖于硬币翻转算子。由傅里叶级数展开导出的初态，本文称之为$(α,β)$离域初态，它提供了与定域初态量子行走不同的极限密度函数。
We construct concrete examples of time operators for both continuous and discrete-time homogeneous quantum walks, and we determine their deficiency indices and spectra. For a discrete-time quantum walk, the time operator can be self-adjoint if the time evolution operator has a non-zero winding number. In this case, its spectrum becomes a discrete set of real numbers.
Absence of zero resonances of massless Dirac operators
本文考虑Hilbert空间$L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$上的无质量Dirac算子$H=α\cdot D+Q(x)$，其中$Q(x)$是一个在无穷远衰减的$4\times4$厄米矩阵值函数.我们证明了$H$不存在零共振，推广了Saito-Umeda和Zhong-Gao最近的结果.
Analytical solutions for quantum walks on 1D chain with different shift operators
本文研究了一维链上具有移动和交换移位算子的离散时间量子行走。利用Chebyshev多项式技术，我们得到了演化算符$\hat{U}$的本征值和本征态的解析解，并分别计算了两种不同的移位算符的长时平均概率.发现移动移位算子和交换移位算子的概率分布具有完全不同的特征。对于移动移位算子，概率分布表现出高度对称性，其中镜像位置处的概率相等。概率与系统大小$N$成反比，并且以$N\rightarrow \infty$的形式接近于零。相反，对于交换移位算子，其概率分布不是对称的，在一定的硬币参数条件下，概率分布趋近于幂律平稳分布$N\rightarrow \infty$.我们表明，这种幂律平稳分布是由本征值$\pm1$的本征态，并计算了不同的起始位置的固有概率。我们的研究结果表明，对应于本征值$\pm1$的本征态对交换移位算子起着重要的作用。
We construct the one-dimensional analogous of von-Neumann Wigner potential to the relativistic Klein-Gordon operator, in which is defined taking asymptotic mathematical rules in order to obtain existence conditions of eigenvalues embedded in the continuous spectrum. Using our constructed potential, we provide an explicit and analytical example of the Klein-Gordon operator with positive eigenvalues embedded in the so called relativistic "continuum region". Even so in this not standard example, we present the region of the "continuum" where those eigenvalues cannot occur. Besides, the absence of…
Dynamical Localization of Quantum Walks in Random Environments
研究了具有两个内部自由度（硬币态）的晶格上一维量子步行者的动力学。离散时间幺正动力学由硬币算子在U（2）中对内部自由度的重复作用决定，然后根据硬币的状态向右或向左移动一步。对于一个固定的硬币运营商，动态是已知的弹道。我们证明了当硬币算子依赖于步行者的位置，并给出了一定的独立同分布。随机过程时，安德森局域化现象以动力学形式发生。当硬币算子仅依赖于时间变量并且由i.i.d.随机过程中，平均运动是已知的扩散，我们计算扩散常数的所有时刻的位置。
The aim of this article is to relate the discrete quantum walk on $\mathbb{Z}$ with the continuous Schrödinger operator on $\mathbb{R}$ in the scattering problem. Each point of $\mathbb{Z}$ is associated with a barrier of the potential, and the coin operator of the quantum walk is determined by the transfer matrix between bases of WKB solutions on the classically allowed regions of both sides of the barrier. This correspondence enables us to represent each entry of the scattering matrix of the Schrödinger operator as a countable sum of probability amplitudes associated with the paths of the qu…
This paper continues the previous work (Quantum Inf. Process (2019)) by two authors of the present paper about a spectral mapping property of chiral symmetric unitary operators. In physics, they treat non-unitary time-evolution operators to consider quantum walks in open systems. In this paper, we generalize the above result to include a chiral symmetric non-unitary operator whose coin operator only has two eigenvalues. As a result, the spectra of such non-unitary operators are included in the (possibly non-unit) circle and the real axis in the complex plane. We also give some examples of our …
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.
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…
Asymptotic entanglement in 1D quantum walks with a time-dependent coined
离散时间量子行走是通过一个幺正算子演化的，其中包含两个算子：一个位置空间中的条件位移算子和一个硬币算子。该操作员纠缠硬币和步行者的位置自由度。本文研究了一维格子中由两个正交矩阵确定的非均匀量子行走中硬币位置纠缠的渐近行为。硬币算符的自由参数共同提供了许多条件，在这些条件下，对硬币态进行测量可以得到所得到的位置量子态上的纠缠值。我们对硬币算符的两个自由参数所能取的所有值解析地研究了这个问题，并寻求了纠缠达到最大的条件。
Absence of flat bands for discrete periodic graph operators with generic potentials
该研究证明了离散连通周期图上的薛定谔型算子对于通用势不存在平带。
We consider an abstract quantum walk defined by a unitary evolution operator $U$, which acts on a Hilbert space decomposed into a direct sum of Hilbert spaces $\{\mathcal{H}_v \}_{v \in V}$. We show that such $U$ naturally defines a directed graph $G_U$ and the probability of finding a quantum walker on $G_U$. The asymptotic property of an abstract quantum walker is governed by the generator $H$ of $U$ such that $U^n = e^{inH}$. We derive the generator of an evolution of the form $U = S(2d_A^* d_A -1)$, a generalization of the Szegedy evolution operator. Here $d_A$ is a boundary operator and $…
Given two Hilbert spaces, $\mathcal{H}$ and $\mathcal{K}$, we introduce an abstract unitary operator $U$ on $\mathcal{H}$ and its discriminant $T$ on $\mathcal{K}$ induced by a coisometry from $\mathcal{H}$ to $\mathcal{K}$ and a unitary involution on $\mathcal{H}$. In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the…
Although quantum walks exhibit peculiar properties that distinguish them from random walks, classical behavior can be recovered in the asymptotic limit by destroying the coherence of the pure state associated to the quantum system. Here I show that this is not the only way: I introduce a quantum walk driven by an inhomogeneous, time-dependent coin operator, which mimics the statistical properties of a random walk at all time scales. The quantum particle undergoes unitary evolution and, in fact, the high correlation evidenced by the components of the wave function can be used to revert the outc…
A Floquet systems is a periodically driven quantum system. It can be described by a Floquet operator. If this unitary operator has a gap in the spectrum, then one can define associated topological bulk invariants which can either only depend on the bands of the Floquet operator or also on the time as a variable. It is shown how a K-theoretic result combined with the bulk-boundary correspondence leads to edge invariants for the half-space Floquet operators. These results also apply to topological quantum walks.
We focus on index theory for chirally symmetric discrete-time quantum walks on the one-dimensional integer lattice. Such a discrete-time quantum walk model can be characterised as a pair of a unitary self-adjoint operator $\varGamma$ and a unitary time-evolution operator $U,$ satisfying the chiral symmetry condition $U^* = \varGamma U \varGamma.$ The significance of this index theory lies in the fact that the index we assign to the pair $(\varGamma,U)$ gives a lower bound for the number of symmetry protected edge-states associated with the time-evolution $U.$ The symmetry protection of edge-st…
We establish concrete criteria for fully supported absolutely continuous spectrum for ergodic CMV matrices and purely absolutely continuous spectrum for limit-periodic CMV matrices. We proceed by proving several variational estimates on the measure of the spectrum and the vanishing set of the Lyapunov exponent for CMV matrices, which represent CMV analogues of results obtained for Schrödinger operators due to Y.\ Last in the early 1990s. Having done so, we combine those estimates with results from inverse spectral theory to obtain purely absolutely continuous spectrum.
We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators $U_0,U$ in Hilbert spaces ${\cal H}_0,{\cal H}$ and for an identification operator $J:{\cal H}_0\to{\cal H}$, we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple $(U,U_0,J)$. In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation form…
We show that the spectral measure of discrete Schrödinger operators $ (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n)$ does not have singular continuous component if the potential $V(n)=O(n^{-1})$.
Measures of Fermi surfaces and absence of singular continuous spectrum for magnetic Schroedinger operators
费米面是固体物理和周期算符谱理论中的基本对象。我们定义了几种与费米面相关的测度，并研究了它们的测度论性质。由此得到了作用在向量丛上的对称周期椭圆微分算子的态密度不存在奇异连续谱和奇异连续分量。这包括具有周期磁场和有理数通量的薛定谔算子，以及相应的泡利和狄拉克型算子。
Absence of eigenvalues for the generalized two-dimensional periodic Dirac operator
考虑了一个广义二维周期Dirac算子，其一阶导数系数为$L^{\infty}$ -矩阵值，具有复矩阵值位势.证明了若矩阵值位势相对于自由Dirac算子有零界，则该算子的谱不含本征值。
Brownian Motion in the $p$-Adic Integers is a Limit of Discrete Time Random Walks
本文证明p进整数环中由Vladimirov-Kochubei算子生成的布朗运动可表示为离散时间随机游走的极限，从而为超度量扩散和有限群上随机过程的弱收敛提供了直观解释。
We show that a generic quasi-periodic Schrödinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely continuous spectrum.
Absence of embedded eigenvalues of Pauli and Dirac operators
本文研究 Pauli 算子嵌入连续谱的特征值，证明在依赖电磁场无穷远渐近行为的阈值之上不存在此类特征值，并说明衰减条件的精确性，结论亦适用于纯磁 Dirac 算子。
Detection of edge defects by embedded eigenvalues of quantum walks
本文研究${\f Z}$上的位置依赖量子游走，提出一种通过时间演化算子的嵌入特征值检测边缘缺陷的方法，并证明在适当假设下有限个边缘缺陷的存在与嵌入特征值的存在等价。
On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials
本文考虑了具有周期势的薛定谔算子在受足够快衰减的"杂质"势扰动时，其连续谱中不存在本征值的问题.这种类型的结果以前只知道一维的情况下。如果相应的费米面是不可约模自然对称，则在二维和三维中不存在嵌入本征值。证明了所有的周期势都满足这个条件。可分周期势满足它，因此在二维和三维空间中，薛定谔算子的可分周期势被一个足够快衰减的"杂质"势扰动，没有嵌入的本征值。
We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators in terms of five integer indices. On the basis of these indices it can be decided whether the half-step operator can be obtained from a continuous Hamiltonian driving, or not. The half-step ope…
Repeated Quantum Interactions and Unitary Random Walks
在描述量子系统$\rH_S$与一系列外部系统重复量子相互作用的离散演化方程中，我们研究并刻画了$\RR^N$中由经典随机变量引导的离散演化方程.我们得到的表征是完全代数的酉算子驱动的基本相互作用。我们证明了这些方程的解是$\rH_0$上酉算子群$U(\rH_0)$上的随机游动.
We perform the scattering analysis of the evolution operator of quantum walks with an anisotropic coin, and we prove a weak limit theorem for their asymptotic velocity. The quantum walks that we consider include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. Our analysis is based on an abstract framework for the scattering theory of unitary operators in a two-Hilbert spaces setting, which is of independent interest.
A weak Gordon type condition for absence of eigenvalues of one-dimensional Schrödinger operators
研究了以复测度为位势的一维Schr dinger算子，提出了一个改进的无本征值判据，该判据包含一个弱的局部周期性条件.该标准导致尖锐的特征值的定量界。我们将所得结果应用于拟周期测度作为势函数。
We study sufficient conditions for the absence of positive eigenvalues of magnetic Schrödinger operators in $\mathbb{R}^d,\, d\geq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller--Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov--Bohm operators.
In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube and then, based on the alternative description, we find that the operators $\partial_k^* + \partial_k$ behave actually as the shift operators, where $\partial_k$ and $\partial_k^*$ are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the above operators as the shift operators on the position space, we introduce a discrete-time quantum walk model on a general hypercube and obtain an explicit for…
We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$. The dispersive estimate $\|U^tP_c u_0\|_{l^\infty}\lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_c$ 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.
Dynamical Localization for d-Dimensional Random Quantum Walks
本文考虑具有随机硬币算符的d维随机量子行走。相应的过渡系数的特点是确定性的振幅时间独立同分布的网站相关的随机相位。当确定性跃迁振幅足够接近禁止传播的量子漫步振幅时，我们证明了动力学局域化对几乎所有随机相都成立。这种安德森局域化的例子意味着位置算符的所有量子力学矩在时间上是一致有界的，并且几乎肯定地说，谱局域化成立。
By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schroedinger operator possesses no point spectrum. The settings of complex-valued electric potentials and singular magnetic potentials of Aharonov-Bohm field are also covered.
We discuss the description of eigenspace of a quantum walk model $U$ with an associating linear operator $T$ in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of $U$ without the spectral decomposition of $T$. Arguments in this direction reveal the eigenspaces of $U$ characterized by the generalized kernels of linear operators given by $T$.
Generalized eigenfunctions and scattering matrices for position-dependent quantum walks
本文研究位置依赖量子游走时间演化算子的谱分析与散射理论，构造了属于$\ell^{\infty}$-空间的广义特征函数，并从时谐散射理论角度刻画其集合，证明S矩阵出现在广义特征函数的奇点展开中。
For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$ and identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we present results on the derivation of a Mourre estimate for $U$ starting from a Mourre estimate for $U_0$ and on the existence and completeness of the wave operators for the triple $(U,U_0,J)$. As an application, we determine spectral and scattering properties of a class of anisotropic quantum walks on homogenous trees of odd degree with evolution operator $U$. In particular, we establish a Mourre estimate for $U$, obtain a class of locally $U$-smooth o…
A remark on the absence of eigenvalues in continuous spectra for discrete Schrödinger operators on periodic lattices
本文在包含正方、三角、六角格点及其梯状结构的一类周期格点上，对具有指数衰减位势的离散薛定谔算子证明 Rellich-Vekua 型定理，并由此讨论唯一延拓性与连续谱中无嵌入特征值。
Localization and Fractality in Inhomogeneous Quantum Walks with Self-Duality
引入并研究了一维格点上的一类离散时间量子行走。与标准的齐次量子行走相反，硬币算子是非齐次的，并且取决于它们在这类模型中的位置。该模型被证明是自对偶相对于傅立叶变换，这是类似于Aubry-André模型描述的一维紧束缚模型与准周期势。当硬币算符的周期与晶格间距不相称时，我们严格地证明了量子行走的极限分布在原点。我们还数值研究了一步时间演化算子的本征值，并发现了Hofstadter蝴蝶谱，这表明了这类量子行走的分形性质。
In this paper we study on $L^2(\mathbb{R}^d)$ the quasi-periodic Schrödinger operator $H=-Δ+ λV(x),$ where $V$ is a real analytic quasi-periodic function and $λ>0$. We first show that $H$ has no eigenvalues in \textit{low energy region}. We also provide in \textit{low energy region} the new phase transition parameter, i.e. the competition between the strength of coupling and the length for frequencies.
On the absence of absolutely continuous spectra for Schrödinger operators on radial tree graphs
本文的主题是薛定谔算子的树图是径向的分支数$b_n$在所有顶点的距离$t_n$从根。我们考虑一族以$(b_n-1)^2+4$实参数为特征的顶点耦合条件。我们证明了，如果图是稀疏的，有一个子序列的$\{t_{n+1}-t_n\}$增长到无穷大，在没有潜在的绝对连续谱是空的一个大的子集，这些顶点耦合，但在另一方面，有情况下，这样的薛定谔算子的谱可以是纯粹绝对连续的。
We prove uniform absence of point spectrum for CMV operators corresponding to the period doubling subshift. We also prove almost sure absence of point spectrum for CMV operators corresponding to a class of Sturmian subshifts. Lastly, we prove almost sure absence of point spectrum for CMV operators corresponding to some subshifts generated by a coding of a rotation.
Poisson operator on the interacting Fock space associated with a discrete-time quantum walk
该论文研究离散时间量子行走对应的相互作用Fock空间上的QW-Poisson算子，分析其谱性质、密度相变，计算矩生成函数等统计量，还关联了非交换概率论。
We address the question of the existence of bound states for a suitably projected two-dimensional massless Dirac operator in the presence of a Bessel-Macdonald potential (also known as $K_0$-potential potential), raised by De Lima, Del Cima and Miranda, in Eur.Phys.J. B (2020) 93, 187. Based on Relativistic Hardy Inequality, we prove that this operator has no bound states if $γ\leqslant γ_{\rm crit}$ (subcritical region), where $γ$ is a coupling constant.
Spectral Property of Magnetic Quantum Walk on Hypercube
本文研究超立方体上磁性量子游走模型，利用量子伯努利噪声构造磁性移位算子，证明演化算子的点谱和近似谱独立于磁势，揭示量子游走对磁场的谱稳定性。
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