Article

Mathematical Physics, Analysis and Geometry

, Volume 12, Issue 2, pp 181-200

Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

  • Maarten BergveltAffiliated withDepartment of Mathematics, University of Illinois
  • , Michael GekhtmanAffiliated withDepartment of Mathematics, University of Notre Dame
  • , Alex KasmanAffiliated withDepartment of Mathematics, College of Charleston Email author 

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Abstract

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.

Keywords

Bispectrality Integrable systems Non-commutative KP hierarchy Spin generalized Calogero-Moser particle system

Mathematics Subject Classifications (2000)

Primary 37K10 Secondary 15A24 34L99 37J35