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Non-periodic one-gap potentials in quantum mechanics. / Zakharov, Dmitry; Zakharov, Vladimir.
Trends in Mathematics. 9783319635934. ред. Springer International Publishing AG, 2018. стр. 221-233 (Trends in Mathematics; № 9783319635934).Результаты исследований: Публикации в книгах, отчётах, сборниках, трудах конференций › глава/раздел › научная › Рецензирование
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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