Mc'Kean's transformation for 3-rd order operators

TL;DR

This paper studies Mc'Kean's transformation that simplifies the spectral analysis of a third order operator related to the good Boussinesq equation by reducing it to a Hill operator with energy-dependent potential.

Contribution

It provides an in-depth analysis of Mc'Kean's transformation for third order operators, elucidating its properties and implications for spectral problems.

Findings

Transformation reduces third order spectral problem to Hill operator

Potential depends analytically on energy

Enhances understanding of spectral properties of third order operators

Abstract

We consider a non-self-adjoint third order operator $(y''+py)'+py'+qy$ with 1-periodic coefficients $p,q$. This operator is the L-operator in the Lax pair for the good Boussinesq equation on the circle. In 1981, McKean introduced a transformation that reduces the spectral problem for this operator to a spectral problem for the Hill operator with a potential that depends analytically on the energy. In the present paper we are studying this transformation.