Non-periodic one-dimensional ideal conductors and integrable turbulence

Dmitry V. Zakharov, Vladimir E. Zakharov, Sergey A. Dyachenko

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

To relate the motion of a quantum particle to the properties of the potential is a fundamental problem of physics, which is far from being solved. Can a medium with a potential which is neither periodic nor quasi-periodic be a conductor? That question seems to have been never addressed, despite being both interesting and having practical importance. Here we propose a new approach to the spectral problem of the one-dimensional Schrödinger operator with a bounded potential. We construct a wide class of potentials having a spectrum consisting of the positive semiaxis and finitely many bands on the negative semiaxis. These potentials, which we call primitive, are reflectionless for positive energy and in general are neither periodic nor quasi-periodic. Moreover, they can be stochastic, and yet allow ballistic transport, and thus describe one-dimensional ideal conductors. Primitive potentials also generate a new class of solutions of the KdV hierarchy. Stochastic primitive potentials describe integrable turbulence, which is important for hydrodynamics and nonlinear optics. We construct the potentials by numerically solving a system of singular integral equations. We hypothesize that finite-gap potentials are a subclass of primitive potentials, and prove this in the case of one-gap potentials.

Original languageEnglish (US)
Pages (from-to)3881-3885
Number of pages5
JournalPhysics Letters, Section A: General, Atomic and Solid State Physics
Volume380
Issue number46
DOIs
StatePublished - Dec 1 2016

Keywords

  • Ideal conductors
  • Integrability
  • Solitonic gas

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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