Mathematical Physics, Analysis and Geometry

, Volume 12, Issue 2, pp 181–200

Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy


  • Maarten Bergvelt
    • Department of MathematicsUniversity of Illinois
  • Michael Gekhtman
    • Department of MathematicsUniversity of Notre Dame
    • Department of MathematicsCollege of Charleston

DOI: 10.1007/s11040-009-9058-y

Cite this article as:
Bergvelt, M., Gekhtman, M. & Kasman, A. Math Phys Anal Geom (2009) 12: 181. doi:10.1007/s11040-009-9058-y


Pairs of n×n matrices whose commutator differ from the identity by a matrix of rank r are used to construct bispectral differential operators with r×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.


BispectralityIntegrable systemsNon-commutative KP hierarchySpin generalized Calogero-Moser particle system

Mathematics Subject Classifications (2000)

Primary 37K10Secondary 15A2434L9937J35

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© Springer Science+Business Media B.V. 2009