Non-periodic one-gap potentials in quantum mechanics

Dmitry Zakharov, Vladimir Zakharov

Research output: Chapter in Book/Report/Conference proceedingChapter


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
Number of pages13
StatePublished - 2018

Publication series

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


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

ASJC Scopus subject areas

  • General Mathematics


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