Kazuyuki Wada
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Kazuyuki Wada has a total of 11 co-authors. Here are the top co-authors by number of shared publications:
- Akito Suzuki
Division of Mathematics and Physics Faculty of Engineering Shinshu University 380-8553 Wakasato Nagano Japan
3 shared publications - Atsuhide Ishida
Katsushika Division Institute of Arts and Sciences Tokyo University of Science 6-3-1 Niijuku Katsushika-ku 125-8585 Tokyo Japan
2 shared publications - Hiromichi Ohno
Department of Mathematics, Faculty of Engineering, Shinshu University, Nagano, Japan
2 shared publications - Masaya Maeda
Department of Mathematics and Informatics, Graduate School of Science, Chiba University, Chiba, Japan
2 shared publications - Akihiro Narimatsu
Graduate School of Science and Engineering, Yokohama National University, Hodogaya, Japan
1 shared publication - Akihiro Narimatsu
Department of Applied Systems and Mathematics, Faculty of Informatics, Kanagawa University, 3-27-1 Rokkakubashi, Yokohama, Kanagawa 221-0802, Japan
1 shared publication
Latest Publications
- 8 June 2024
- Published by Springer Nature in Advances in Operator Theory
- Vol. 9 (3) , 1-13
Exponential decay property for eigenfunctions of quantum walks
- 8 June 2024
- Published by Springer Nature in Advances in Operator Theory
- Vol. 9 (3) , 1-13
Exponential decay property for eigenfunctions of quantum walks
- Open Access
The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition
It is recently shown that a split-step quantum walk possesses a chiral symmetry, and that a certain well-defined index can be naturally assigned to it. The index is a well-defined Fredholm index if and only if the associated unitary time-evolution operator has spectral gaps at both $+1$ and $-1.$ In this paper we extend the existing index formula for the Fredholm case to encompass the non-Fredholm case (i.e., gapless case). We make use of a natural extension of the Fredholm index to the non-Fredholm case, known as the Witten index. The aim of this paper is to fully classify the Witten index of the split-step quantum walk by employing the spectral shift function for a rank one perturbation of a fourth order difference operator. It is also shown in this paper that the Witten index can take half-integer values in the non-Fredholm case.Open AccessThe Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition
It is recently shown that a split-step quantum walk possesses a chiral symmetry, and that a certain well-defined index can be naturally assigned to it. The index is a well-defined Fredholm index if and only if the associated unitary time-evolution operator has spectral gaps at both $+1$ and $-1.$ In this paper we extend the existing index formula for the Fredholm case to encompass the non-Fredholm case (i.e., gapless case). We make use of a natural extension of the Fredholm index to the non-Fredholm case, known as the Witten index. The aim of this paper is to fully classify the Witten index of the split-step quantum walk by employing the spectral shift function for a rank one perturbation of a fourth order difference operator. It is also shown in this paper that the Witten index can take half-integer values in the non-Fredholm case. - Open Access
- 31 March 2022
- Published by World Scientific Pub Co Pte Ltd in Reviews in Mathematical Physics
- Vol. 34 (05)
Absence of singular continuous spectra and embedded eigenvalues for one-dimensional quantum walks with general long-range coins
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.
- 31 March 2022
- Published by World Scientific Pub Co Pte Ltd in Reviews in Mathematical Physics
- Vol. 34 (05)
Open AccessAbsence of singular continuous spectra and embedded eigenvalues for one-dimensional quantum walks with general long-range coins
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.
- Open Access
Index Theorems for One-dimensional Chirally Symmetric Quantum Walks with Asymptotically Periodic Parameters
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-states is one of the important features of the bulk-edge correspondence. The purpose of the present paper is to revisit the well-known bulk-edge correspondence for the split-step quantum walk on the one-dimensional integer lattice. The existing mathematics literature makes use of a fundamental assumption, known as the $2$-phase condition, but we completely replace it by the so-called asymptotically periodic assumption in this article. This generalisation heavily relies on analysis of some topological invariants associated with Toeplitz operators.Open AccessIndex Theorems for One-dimensional Chirally Symmetric Quantum Walks with Asymptotically Periodic Parameters
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-states is one of the important features of the bulk-edge correspondence. The purpose of the present paper is to revisit the well-known bulk-edge correspondence for the split-step quantum walk on the one-dimensional integer lattice. The existing mathematics literature makes use of a fundamental assumption, known as the $2$-phase condition, but we completely replace it by the so-called asymptotically periodic assumption in this article. This generalisation heavily relies on analysis of some topological invariants associated with Toeplitz operators. - 5 November 2021
- Published by Springer Nature in Quantum Information Processing
- Vol. 20 (11) , 1-23
Unitary equivalence classes of split-step quantum walks
- 5 November 2021
- Published by Springer Nature in Quantum Information Processing
- Vol. 20 (11) , 1-23
Unitary equivalence classes of split-step quantum walks
- Open Access
Absence of singular continuous spectra and embedded eigenvalues for one dimensional quantum walks with general long-range coins
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}.
Open AccessAbsence of singular continuous spectra and embedded eigenvalues for one dimensional quantum walks with general long-range coins
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}.
- Open Access
Unitary equivalence classes of split-step quantum walks
This study investigates the unitary equivalence of split-step quantum walks (SSQW). We consider a new class of quantum walks which includes all SSQWs. We show the explicit form of quantum walks in this class, and clarify their unitary equivalence classes. Unitary equivalence classes of Suzuki's SSQW are also given.
Open AccessUnitary equivalence classes of split-step quantum walks
This study investigates the unitary equivalence of split-step quantum walks (SSQW). We consider a new class of quantum walks which includes all SSQWs. We show the explicit form of quantum walks in this class, and clarify their unitary equivalence classes. Unitary equivalence classes of Suzuki's SSQW are also given.
- 25 August 2020
- Published by Springer Nature in Mathematical Physics, Analysis and Geometry
- Vol. 23 (3) , 1-11
Threshold Between Short and Long-range Potentials for Non-local Schrödinger Operators
- 25 August 2020
- Published by Springer Nature in Mathematical Physics, Analysis and Geometry
- Vol. 23 (3) , 1-11
Threshold Between Short and Long-range Potentials for Non-local Schrödinger Operators
- Open Access
Threshold between short and long-range potentials for non-local Schrödinger operators
We develop scattering theory for non-local Schr\"odinger operators defined by functions of the Laplacian that include its fractional power $(-\Delta)^\rho$ with $0<\rho\leqslant1$. In particular, our function belongs to a wider class than the set of Bernstein functions. By showing the existence and non-existence of the wave operators, we clarify the threshold between the short and long-range decay conditions for perturbational potentials.Open AccessThreshold between short and long-range potentials for non-local Schrödinger operators
We develop scattering theory for non-local Schr\"odinger operators defined by functions of the Laplacian that include its fractional power $(-\Delta)^\rho$ with $0<\rho\leqslant1$. In particular, our function belongs to a wider class than the set of Bernstein functions. By showing the existence and non-existence of the wave operators, we clarify the threshold between the short and long-range decay conditions for perturbational potentials. - 13 November 2019
- Published by Springer Nature in Quantum Information Processing
- Vol. 19 (1) , 1-17
A weak limit theorem for a class of long-range-type quantum walks in 1d
- 13 November 2019
- Published by Springer Nature in Quantum Information Processing
- Vol. 19 (1) , 1-17
A weak limit theorem for a class of long-range-type quantum walks in 1d