Self-adjoint commuting differential operators and commutative subalgebras of the Weyl algebra

Abstract

In this paper we study self-adjoint commuting ordinary differential operators. We find sufficient conditions when an operator of fourth order commuting with an operator of order $4g+2$ is self-adjoint. We introduce an equation on coefficients of the self-adjoint operator of order four and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of arbitrary genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.

References in corpus (2)

• On Darboux-Treibich-Verdier potentials
• Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2

Cited by in corpus (7)

• Commuting differential operators of rank 2 with polynomial coefficients
• Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2
• Common eigenfunctions of commuting differential operators of rank 2
• On rings of commuting partial differential operators
• Matrix commuting differential operators of rank 2
• AKNS hierarchy and finite-gap Schrodinger potentials
• Commuting differential operators of rank 2 with rational coefficients