Non-periodic one-gap potentials in quantum mechanics

Dmitry Zakharov, Vladimir E Zakharov

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

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.

Original languageEnglish (US)
Title of host publicationTrends in Mathematics
PublisherSpringer International Publishing
Pages221-233
Number of pages13
Edition9783319635934
DOIs
StatePublished - Jan 1 2018

Publication series

NameTrends in Mathematics
Number9783319635934
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Keywords

  • Integrable turbulence
  • KdV equation
  • Riemann–Hilbert problem
  • Schrödinger operator

ASJC Scopus subject areas

  • Mathematics(all)

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