Scattering and inverse scattering for nonlinear quantum walks

Masaya Maeda; Hironobu Sasaki; Etsuo Segawa; Akito Suzuki; Kanako Suzuki

doi:10.3934/dcds.2018159

Scattering and inverse scattering for nonlinear quantum walks

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki

Information Sciences

Research output: Contribution to journal › Article › peer-review

15 Citations (Scopus)

Abstract

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 QWs.

Original language	English
Pages (from-to)	3687-3703
Number of pages	17
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	38
Issue number	7
DOIs	https://doi.org/10.3934/dcds.2018159
Publication status	Published - 2018 Jul

Keywords

Dispersive estimates
Nonlinear scattering.
Quantum walks
Scattering theory
Strichartz estimates

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2018159

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title = "Scattering and inverse scattering for nonlinear quantum walks",

abstract = "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 Schr{\"o}dinger equations and discrete nonlinear Schr{\"o}dinger equations but it seems to be the first time to be applied to QWs.",

keywords = "Dispersive estimates, Nonlinear scattering., Quantum walks, Scattering theory, Strichartz estimates",

author = "Masaya Maeda and Hironobu Sasaki and Etsuo Segawa and Akito Suzuki and Kanako Suzuki",

note = "Funding Information: M.M. was supported by the JSPS KAKENHI Grant Numbers JP15K17568, JP17H02851 and JP17H02853. H.S. was supported by JSPS KAKENHI Grant Number JP17K05311. E.S. acknowledges financial support from the Grant-in-Aid for Young Scientists (B) and of Scientific Research (B) Japan Society for the Promotion of Science (Grant No. 16K17637, No. 16K03939). A. S. was supported by JSPS KAKENHI Grant Number JP26800054. K.S acknowledges JSPS the Grant-in-Aid for Scientific Research (C) 26400156. We thank Professor Norio Konno for many valuable advices. Publisher Copyright: {\textcopyright} 2018 American Institute of Mathematical Sciences. All rights reserved.",

year = "2018",

month = jul,

doi = "10.3934/dcds.2018159",

language = "English",

volume = "38",

pages = "3687--3703",

journal = "Discrete and Continuous Dynamical Systems",

issn = "1078-0947",

publisher = "Southwest Missouri State University",

number = "7",

}

TY - JOUR

T1 - Scattering and inverse scattering for nonlinear quantum walks

AU - Maeda, Masaya

AU - Sasaki, Hironobu

AU - Segawa, Etsuo

AU - Suzuki, Akito

AU - Suzuki, Kanako

N1 - Funding Information: M.M. was supported by the JSPS KAKENHI Grant Numbers JP15K17568, JP17H02851 and JP17H02853. H.S. was supported by JSPS KAKENHI Grant Number JP17K05311. E.S. acknowledges financial support from the Grant-in-Aid for Young Scientists (B) and of Scientific Research (B) Japan Society for the Promotion of Science (Grant No. 16K17637, No. 16K03939). A. S. was supported by JSPS KAKENHI Grant Number JP26800054. K.S acknowledges JSPS the Grant-in-Aid for Scientific Research (C) 26400156. We thank Professor Norio Konno for many valuable advices. Publisher Copyright: © 2018 American Institute of Mathematical Sciences. All rights reserved.

PY - 2018/7

Y1 - 2018/7

N2 - 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 QWs.

AB - 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 QWs.

KW - Dispersive estimates

KW - Nonlinear scattering.

KW - Quantum walks

KW - Scattering theory

KW - Strichartz estimates

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DO - 10.3934/dcds.2018159

M3 - Article

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VL - 38

SP - 3687

EP - 3703

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 7

ER -

Scattering and inverse scattering for nonlinear quantum walks

Abstract

Keywords

ASJC Scopus subject areas

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