TY - JOUR

T1 - Non-periodic one-dimensional ideal conductors and integrable turbulence

AU - Zakharov, Dmitry V.

AU - Zakharov, Vladimir E.

AU - Dyachenko, Sergey A.

N1 - Funding Information:
The authors would like to thank Harry Braden, Percy Deift, Igor Krichever and Thomas Trogdon for insightful discussions. The authors gratefully acknowledge the support of Russian Science Foundation grant number 14-22-00174 for analytic work and developing numerical simulations.
Publisher Copyright:
© 2016 Elsevier B.V.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

KW - Ideal conductors

KW - Integrability

KW - Solitonic gas

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U2 - 10.1016/j.physleta.2016.09.040

DO - 10.1016/j.physleta.2016.09.040

M3 - Article

AN - SCOPUS:84991823638

VL - 380

SP - 3881

EP - 3885

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

SN - 0375-9601

IS - 46

ER -