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Non-periodic one-gap potentials in quantum mechanics. / Zakharov, Dmitry; Zakharov, Vladimir.

Trends in Mathematics. 9783319635934. ed. Springer International Publishing AG, 2018. p. 221-233 (Trends in Mathematics; No. 9783319635934).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Research › peer-review

Harvard

Zakharov, D & Zakharov, V 2018, Non-periodic one-gap potentials in quantum mechanics. in Trends in Mathematics. 9783319635934 edn, Trends in Mathematics, no. 9783319635934, Springer International Publishing AG, pp. 221-233. https://doi.org/10.1007/978-3-319-63594-1_22

APA

Zakharov, D., & Zakharov, V. (2018). Non-periodic one-gap potentials in quantum mechanics. In Trends in Mathematics (9783319635934 ed., pp. 221-233). (Trends in Mathematics; No. 9783319635934). Springer International Publishing AG. https://doi.org/10.1007/978-3-319-63594-1_22

Vancouver

Zakharov D, Zakharov V. Non-periodic one-gap potentials in quantum mechanics. In Trends in Mathematics. 9783319635934 ed. Springer International Publishing AG. 2018. p. 221-233. (Trends in Mathematics; 9783319635934). doi: 10.1007/978-3-319-63594-1_22

Author

Zakharov, Dmitry ; Zakharov, Vladimir. / Non-periodic one-gap potentials in quantum mechanics. Trends in Mathematics. 9783319635934. ed. Springer International Publishing AG, 2018. pp. 221-233 (Trends in Mathematics; 9783319635934).

BibTeX

@inbook{9f1a8420b96c4cd78b946fcdbe2144c7,
title = "Non-periodic one-gap potentials in quantum mechanics",
abstract = "We construct a broad class of bounded potentials of the one-dimensional Schr{\"o}dinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive H{\"o}lder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.",
keywords = "Integrable turbulence, KdV equation, Riemann–Hilbert problem, Schr{\"o}dinger operator",
author = "Dmitry Zakharov and Vladimir Zakharov",
year = "2018",
month = jan,
day = "1",
doi = "10.1007/978-3-319-63594-1_22",
language = "English",
series = "Trends in Mathematics",
publisher = "Springer International Publishing AG",
number = "9783319635934",
pages = "221--233",
booktitle = "Trends in Mathematics",
address = "Switzerland",
edition = "9783319635934",

}

RIS

TY - CHAP

T1 - Non-periodic one-gap potentials in quantum mechanics

AU - Zakharov, Dmitry

AU - Zakharov, Vladimir

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We construct a broad class of bounded potentials of the one-dimensional Schrödinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.

AB - We construct a broad class of bounded potentials of the one-dimensional Schrödinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.

KW - Integrable turbulence

KW - KdV equation

KW - Riemann–Hilbert problem

KW - Schrödinger operator

UR - http://www.scopus.com/inward/record.url?scp=85041961444&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-63594-1_22

DO - 10.1007/978-3-319-63594-1_22

M3 - Chapter

AN - SCOPUS:85041961444

T3 - Trends in Mathematics

SP - 221

EP - 233

BT - Trends in Mathematics

PB - Springer International Publishing AG

ER -

ID: 10421163