Third order operator with periodic coefficients on the real line

TL;DR

This paper analyzes a third order periodic differential operator on the real line, establishing its spectral properties, self-adjointness, and the structure of its spectrum, with implications for the Boussinesq equation.

Contribution

It proves the self-adjointness, spectral decomposition, and constructs the Lyapunov function for the third order operator with minimal smoothness of coefficients.

Findings

Operator is self-adjoint and decomposable into a direct integral.

Spectrum is absolutely continuous, covers the entire real line, with multiplicity one or three.

Constructed and analyzed the Lyapunov function on a three-sheeted Riemann surface.

Abstract

We consider the third order operator with periodic coefficients on the real line. This operator is used in the integration of the non-linear evolution Boussinesq equation. For the minimal smoothness of the coefficients we prove that: 1) the operator is self-adjoint and it is decomposable into the direct integral, 2) the spectrum is absolutely continuous, fills the whole real axis, and has multiplicity one or three, 3) the Lyapunov function, analytic on a three-sheeted Riemann surface, is constructed and researched, 4) the spectrum of multiplicity three is bounded and it is described in terms of some entire function (the discriminant).