Nonperiodic onedimensional ideal conductors and integrable turbulence
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 quasiperiodic 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 onedimensional 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 quasiperiodic. Moreover, they can be stochastic, and yet allow ballistic transport, and thus describe onedimensional 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 finitegap potentials are a subclass of primitive potentials, and prove this in the case of onegap potentials.
 Publication:

Physics Letters A
 Pub Date:
 December 2016
 DOI:
 10.1016/j.physleta.2016.09.040
 Bibcode:
 2016PhLA..380.3881Z
 Keywords:

 Ideal conductors;
 Integrability;
 Solitonic gas