Vladimir Zakharov

Schrodinger operator with continuous spectrum and turbulence in integrable systems

We study one-dimensional stationary Schrodinger operator with bounded on the line non-periodic and non-quasiperiodic potentials. We address the following question: Could the Schrodinger operator have a spectrum similar to the spectrum of an operator with periodic potential, pure continuous spectrum of band type? The answer is positive. We developed an algorithm for construction of such potentials based on solution of certain non-local Riemann-Hilbert problem on the plane of complex wave numbers. The wave function of such operator is an analytic function on this plane with exception of symmetry posed cuts. In a general case the constructed potentials are random, they describe random bounded solutions of the Korteweg - de Vries (KdV) equation. Temporal evolution of such potentials can be interpreted as integrable turbulence in framework of the KdV equation. Analytic results are supported by numerical experiments.

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