Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems

TL;DR

This paper introduces a new approach using dual grassmannians to analyze rational and soliton solutions of the KP hierarchy, revealing that bispectrality relates to bounded tau function degrees and linearizes Calogero-Moser systems.

Contribution

It develops a novel construction with dual grassmannians for KP solutions and explicitly characterizes the bispectral involution as a linearizing map for Calogero-Moser particles.

Findings

Bound on tau function degree iff wave function is rank one and bispectral

Explicit description of bispectral involution in terms of grassmannian parameters

Bispectral involution acts as a linearizing map for Calogero-Moser systems

Abstract

A new construction using finite dimensional dual grassmannians is developed to study rational and soliton solutions of the KP hierarchy. In the rational case, properties of the tau function which are equivalent to bispectrality of the associated wave function are identified. In particular, it is shown that there exists a bound on the degree of all time variables in tau if and only if the wave function is rank one and bispectral. The action of the bispectral involution, beta, in the generic rational case is determined explicitly in terms of dual grassmannian parameters. Using the correspondence between rational solutions and particle systems, it is demonstrated that beta is a linearizing map of the Calogero-Moser particle system and is essentially the map sigma introduced by Airault, McKean and Moser in 1977.