arXiv subjects

Akito Suzuki

Publications and source records attributed to Akito Suzuki.

34 records · Page 2Linked to original sources

arXiv2018

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 are robust against perturbations.

math-ph↗

arXiv2018

Scattering and inverse scattering for nonlinear quantum walks

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations and discrete nonlinear Schrödinger equations but it seems to be the first time to be applied to QW.

math-ph↗

arXiv2018

Weak limit theorem for a nonlinear quantum walk

This paper continues the study of large time behavior of a nonlinear quantum walk begun in arXiv:1801.03214. In this paper, we provide a weak limit theorem for the distribution of the nonlinear quantum walk. The proof is based on the scattering theory of the nonlinear quantum walk and the limit distribution is obtained in terms of its asymptotic state.

math-ph↗

arXiv2017

On nonlinear scattering for quantum walks

We study large time behavior of quantum walks (QW) with self-dependent coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QW such as Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations but it seems to be the first time to be applied to QW. We also numerically study the dynamics of QW and observe soliton like solutions.

math-ph↗

arXiv2017

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.

math-ph↗

arXiv2016

Spectral mapping theorem of an abstract quantum walk

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 Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.

math-ph↗

arXiv2016

Asymptotic velocity of a position-dependent quantum walk

We consider a position-dependent coined quantum walk on $\mathbb{Z}$ and assume that the coin operator $C(x)$ satisfies \[ \|C(x) - C_0 \| \leq c_1|x|^{-1-ε}, \quad x \in \mathbb{Z} \] with positive $c_1$ and $ε$ and $C_0 \in U(2)$. We show that the Heisenberg operator $\hat x(t)$ of the position operator converges to the asymptotic velocity operator $\hat v_+$ so that \[ \mbox{s-}\lim_{t \to \infty} {\rm exp}\left( i ξ\frac{\hat x(t)}{t} \right) = Π_{\rm p}(U) + {\rm exp}(i ξ\hat v_+) Π_{\rm ac}(U) \] provided that $U$ has no singular continuous spectrum. Here $Π_{\rm p}(U)$ (resp. $Π_{\rm ac}(U)$) is the orthogonal projection onto the direct sum of all eigenspaces (resp. the subspace of absolute continuity) of $U$. We also prove that for the random variable $X_t$ denoting the position of a quantum walker at time $t \in \mathbb{N}$, $X_t/t$ converges in law to a random variable $V$ with the probability distribution \[ μ_V = \|Π_{\rm p}(U)Ψ_0\|^2δ_0 + \|E_{\hat v_+}(\cdot) Π_{\rm ac}(U)Ψ_0\|^2, \] where $Ψ_0$ is the initial state, $δ_0$ the Dirac measure at zero, and $E_{\hat v_+}$ the spectral measure of $\hat v_+$.

math.SP↗

arXiv2016

Generator of an abstract quantum walk

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 $S$ a shift operator.

math-ph↗

arXiv2015

Non-commutative hypergroup of order five

We prove that all hypergroups of order four are commutative and that there exists a non-comutative hypergroup of order five. These facts imply that the minimum order of non-commutative hypergroups is five even though the minimum order of non-commutative groups is six.

math.GR↗

arXiv2015

Essential spectrum of the discrete Laplacian on a perturbed periodic graph

We address the Laplacian on a perturbed periodic graph which might not be a periodic graph. We present a class of perturbed graphs for which the essential spectra of the Laplacians are stable even when the graphs are perturbed by adding and removing infinitely many vertices and edges. Using this result, we demonstrate how to determine the spectra of cone-like graphs, the upper-half plane, and graphs obtained from $\mathbb{Z}^2$ by randomly adding vertices.

math-ph↗

arXiv2012

Note on the spectrum of discrete Schrödinger operators

The spectrum of discrete Schrödinger operator $L+V$ on the $d$-dimensional lattice is considered, where $L$ denotes the discrete Laplacian and $V$ a delta function with mass at a single point. Eigenvalues of $L+V$ are specified and the absence of singular continuous spectrum is proven. In particular it is shown that an embedded eigenvalue does appear for $d\geq5$ but does not for $1\leq d\leq 4$.

math-ph↗

arXiv2012

Enhanced Binding in Quantum Field Theory

This lecture note consists of three parts. Fundamental facts on Boson Fock space are introduced in Part I. Ref. 1.and 3. are reviewed in Part II and, Ref. 2. and 4. in Part III. In Part I a symplectic structure of a Boson Fock space is studied and a projective unitary representation of an infinite dimensional symplectic group through Bogoliubov transformations is constructed. In Part II the so-called Pauli-Fierz model (PF model) with the dipole approximation in non-relativistic quantum electrodynamics is investigated. This model describes a minimal interaction between a massless quantized radiation field and a quantum mechanical particle (electron) governed by Schrödinger operator. By applying the Bogoliubov transformation introduced in Part I we investigate the spectrum of the PF model. First the translation invariant case is considered and the dressed electron state with a fixed momentum is studied. Secondly the absence of ground state is proven by extending the Birman-Schwinger principle. Finally the enhanced binding of a ground state is discussed and the transition from unbinding to binding is shown. In Part III the so-called $N$-body Nelson model is studied. This model describes a linear interaction between a scalar field and $N$-body quantum mechanical particles. First the enhanced binding is shown by checking the so-called stability condition. Secondly the Nelson model with variable coefficients is discussed, which model can be derived when the Minkowskian space-time is replaced by a static Riemannian manifold, and the absence of ground state is proven, if the variable mass decays to zero sufficiently fast. The strategy is based on a path measure argument.

math-ph↗

arXiv2011

The No-Binding Regime of the Pauli-Fierz Model

The Pauli-Fierz model $H(α)$ in nonrelativistic quantum electrodynamics is considered. The external potential $V$ is sufficiently shallow and the dipole approximation is assumed. It is proven that there exist constants $0<α_-< α_+$ such that $H(α)$ has no ground state for $|α|<α_-$, which complements an earlier result stating that there is a ground state for $|α| > α_+$. We develop a suitable extension of the Birman-Schwinger argument. Moreover for any given $δ>0$ examples of potentials $V$ are provided such that $α_+-α_-<δ$.

math-ph↗

arXiv2011

Infrared problem for the Nelson model on static space-times

We consider the Nelson model with variable coefficients and investigate the problem of existence of a ground state and the removal of the ultraviolet cutoff. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. A physical example is obtained by quantizing the Klein-Gordon equation on a static space-time coupled with a non-relativistic particle. We investigate the existence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass tends to 0 at infinity.

math-ph↗

arXiv2011

An inverse scattering problem for the Klein-Gordon equation with a classical source in quantum field theory

An inverse scattering problem for a quantized scalar field ${\bm ϕ}$ obeying a linear Klein-Gordon equation $(\square + m^2 + V) {\bm ϕ} = J \mbox{in $\mathbb{R} \times \mathbb{R}^3$}$ is considered, where $V$ is a repulsive external potential and $J$ an external source $J$. We prove that the scattering operator $\mathscr{S}= \mathscr{S}(V,J)$ associated with ${\bm ϕ}$ uniquely determines $V$. Assuming that $J$ is of the form $J(t,x)=j(t)ρ(x)$, $(t,x) \in \mathbb{R} \times \mathbb{R}^3$, we represent $ρ$ (resp. $j$) in terms of $j$ (resp. $ρ$) and $\mathscr{S}$.

math-ph↗

arXiv2008

Physical state for non-relativistic quantum electrodynamics

A physical subspace and physical Hilbert space associated with asymptotic fields of nonrelativistic quantum electrodynamics are constructed through the Gupta-Bleuler procedure. Asymptotic completeness is shown and a physical Hamiltonian is defined on the physical Hilbert space.

math-ph↗