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Daiju Funakawa

Publications and source records attributed to Daiju Funakawa.

11 recordsLinked to original sources
arXiv2026

Poisson operator on the interacting Fock space associated with a discrete-time quantum walk

We study the Poisson operator on the interacting Fock space associated with a discrete-time quantum walk, which we call the QW-Poisson operator. First, we investigate the spectral properties of the QW-Poisson distribution. In particular, we establish a relation between the spectral distributions of the Poisson operator and the reversed Poisson operator on a general interacting Fock space via a size-biased transform. Next, we study the edge behavior of the density of the QW-Poisson distribution. We show that a phase transition occurs at the left endpoint of the support: depending on the parameter, the density either decays to $0$ or blows up to $+\infty$. Moreover, this phase transition coincides with the transition in the number of atoms of the QW-Poisson distribution, equivalently, in the point spectrum of the QW-Poisson operator, and with whether $0$ belongs to the spectrum of the QW-Poisson operator. Finally, we study a connection between the interacting Fock space associated with a discrete-time quantum walk and noncommutative probability theory. More precisely, we compute the moment-generating function and moments of the QW-Poisson operator, and obtain a limit theorem for the Konno distribution via a Poisson approximation. We also investigate the Boolean self-decomposability of the Konno distribution and the shifted reversed QW-Poisson distribution.

math.FA↗
arXiv2025

Two-dimensional quantum central limit theorem by quantum walks

The weak limit theorem (WLT), the quantum analogue of the central limit theorem, is foundational to quantum walk (QW) theory. Unlike the universal Gaussian limit of classical walks, deriving analytical forms of the limiting probability density function (PDF) in higher dimensions has remained a challenge since the 1D Konno distribution was established. Previous explicit PDFs for 2D models were limited to specific cases whose fundamental nature was unclear. This paper resolves this long-standing gap by introducing the notion of maximal speed $v_{\mathrm{max}}$ as a critical parameter. We demonstrate that all previous 2D solutions correspond to a degenerate regime where $v_{\mathrm{max}} = 1$. We then present the first exact analytical representation of the limiting PDF for the physically richer, unexplored regime $v_{\mathrm{max}} < 1$ of a general class of 2D two-state QWs. Our result reveals 2D Konno functions that govern these dynamics. We establish these as the proper 2D generalization of the 1D Konno distribution by demonstrating their convergence to the 1D form in the appropriate limit. Furthermore, our derivation, based on spectral analysis of the group velocity map, analytically resolves the singular asymptotic structure: we explicitly determine the caustics loci where the PDF diverges and prove they define the boundaries of the distribution's support. By also providing a closed-form expression for the weight functions, this work offers a complete description of the 2D WLT.

math-ph↗
arXiv2023

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

In this paper, we derive sufficient conditions for the localization of two-dimensional split-step quantum walks with a strong shift. For this purpose, we analyze the zero points of the function $f$ introduced by Fuda et. al. (Quantum Inf Process 16(8) 203, 2017) and make these zero points explicit. These zeros provide a concrete representation of the eigenvalues and eigenvectors of the evolution operator, and in particular, clarify where localization occurs. In addition, the eigenvalues obtained here asymptotically approach threshold resonance in special cases. We also describe the display of threshold resonances and generalized eigenfunctions.

math-ph↗
arXiv2022

Spectral mapping theorem of an abstract non-unitary quantum walk

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 abstract results, such as non-unitary quantum walks defined by Mochizuki et al. Moreover, we present an application to the Ihara zeta functions and correlated random walks on regular graphs, which are not quantum walks.

math-ph↗
arXiv2020

The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution

Recent developments in the index theory of discrete-time quantum walks allow us to assign a certain well-defined supersymmetric index to a pair of a unitary time-evolution $U$ and a $\mathbb{Z}_2$-grading operator $\varGamma$ satisfying the chiral symmetry condition $U^* = \varGamma U \varGamma.$ In this paper, this index theory will be extended to encompass non-unitary $U$. The existing literature for unitary $U$ makes use of the indispensable assumption that $U$ is essentially gapped; that is, we require that the essential spectrum of $U$ contains neither $-1$ nor $+1$ to define the associated index. It turns out that this assumption is no longer necessary, if the given time-evolution $U$ is non-unitary. As a concrete example, we shall consider a well-known non-unitary quantum walk model on the one-dimensional integer lattice, introduced by Mochizuki-Kim-Obuse.

math-ph↗
arXiv2020

Time operators for continuous-time and discrete-time quantum walks

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.

math-ph↗
arXiv2018

Spectral analysis of an abstract pair interaction model

We consider an abstract pair-interaction model in quantum field theory with a coupling constant $λ\in {\mathbb R}$ and analyze the Hamiltonian $H(λ)$ of the model. In the massive case, there exist constants $λ_{\rm c}<0$ and $λ_{{\rm c},0}<λ_{\rm c}$ such that, for each $λ\in (λ_{{\rm c},0},λ_{\rm c})\cup (λ_{\rm c},\infty)$, $H(λ)$ is diagonalized by a proper Bogoliubov transformation, so that the spectrum of $H(λ)$ is explicitly identified, where the spectrum of $H(λ)$ for $λ>λ_{\rm c}$ is different from that for $λ\in (λ_{{\rm c},0}, λ_{\rm c})$. As for the case $λ<λ_{{\rm c},0}$, we show that $H(λ)$ is unbounded from above and below. In the massless case, $λ_{\rm c}$ coincides with $λ_{{\rm c},0}$.

math-ph↗
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↗
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↗