Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

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.

16 pages

References in corpus (4)

• Physics and Mathematics of Calogero particles
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• On KP Generators and the Geometry of the HBDE
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Cited by in corpus (8)

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• Time-dependent Schroedinger equation in dimension $k+1$: explicit and rational solutions via GBDT and multinodes
• Bispectrality of $N$-Component KP Wave Functions: A Study in Non-Commutativity
• Pseudo-Exponential-Type Solutions of Wave Equations Depending on Several Variables
• Exceptional Hermite Polynomials and Calogero-Moser Pairs
• The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation