Periodic and rapid decay rank two self-adjoint commuting differential operators

TL;DR

This paper constructs self-adjoint rank two commuting differential operators with various coefficients linked to hyperelliptic spectral curves, and discusses related problems in Lame operators and soliton equations.

Contribution

It introduces new constructions of self-adjoint rank two commuting differential operators with trigonometric, elliptic, and rapid decay coefficients.

Findings

Operators with hyperelliptic spectral curves are explicitly constructed.

Connections to Lame operators and soliton solutions are analyzed.

The work advances understanding of rank two commuting differential operators.

Abstract

Self-adjoint rank two commuting ordinary differential operators are studied in this paper. Such operators with trigonometric, elliptic and rapid decay coefficients corresponding to hyperelliptic spectral curves are constructed. Some problems related to the Lame operator and rank two solutions of soliton equations are discussed.