Non-periodic one-gap potentials in quantum mechanics

Dmitry Zakharov, Vladimir Zakharov

Результат исследования: Публикации в книгах, отчётах, сборниках, трудах конференцийглава/разделнаучнаярецензирование


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.

Язык оригиналаанглийский
Название основной публикацииTrends in Mathematics
ИздательSpringer International Publishing AG
Число страниц13
СостояниеОпубликовано - 1 янв 2018

Серия публикаций

НазваниеTrends in Mathematics
ISSN (печатное издание)2297-0215
ISSN (электронное издание)2297-024X

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    Zakharov, D., & Zakharov, V. (2018). Non-periodic one-gap potentials in quantum mechanics. В Trends in Mathematics (9783319635934 ред., стр. 221-233). (Trends in Mathematics; № 9783319635934). Springer International Publishing AG.