Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

TL;DR

This paper constructs bispectral matrix differential operators linked to spin Calogero particles using matrix pairs with specific commutator properties, extending known scalar bispectral results to a matrix setting.

Contribution

It introduces a new class of bispectral matrix differential operators associated with spin Calogero particles, generalizing previous scalar bispectrality results to matrix operators.

Findings

Constructed bispectral operators with matrix coefficients satisfying Matrix KP hierarchy.

Linked bispectral involution to dynamics of spin Calogero particles.

Extended scalar bispectrality results to matrix differential operators.

Abstract

Pairs of $n\times n$ matrices whose commutator differ from the identity by a matrix of rank $r$ are used to construct bispectral differential operators with $r\times r$ matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case $r=1$, this reproduces well-known results of Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators. This new class of pairs $(L, \Lambda)$ of bispectral matrix differential operators is different than those previously studied in that $L$ acts from the left, but $\Lambda$ from the right on a common $r\times r$ eigenmatrix.