Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

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. This new class of pairs (L, Λ) of bispectral matrix differential operators is different than those previously studied in that L acts from the left, but Λ from the right on a common r × r eigenmatrix.

Time discretization of the spin Calogero-Moser model and the semi-discrete matrix KP hierarchy

“…Here we show that the linear problems (24), (25), which are the basic tools for deriving the equations of motion for the discrete time pole dynamics, are equivalent to corollaries of the bilinear identity (12). The derivation is similar to the one given in [27].…”

“…Here we list some corollaries of the basic bilinear identity (12) which are used below in the appendix for the derivation of the linear problems (24), (25).…”

“…Setting t ′ = t + [µ −1 ] − [ν −1 ] α , we represent the bilinear identity (12) at β = α in the form…”

KP hierarchy for the cyclic quiver

“…Its relation with the matrix KP hierarchy and the spaces C n,λ has been considered in [KBBT], [W2], [BeGeKa], [BP]. Note that even in the m = 1 case, our Theorem 6.5 gives a little bit more compared to [BeGeKa,Theorem 3.5], since we consider the full multicomponent KP hierarchy, while in [BeGeKa] only the flows corresponding to ∂ t k = d β=1 ∂ t k,β are considered. 6.4.…”

Integral operators, bispectrality and growth of Fourier algebras

“…Therefore for bisymmetry with respect to (Γ 1 , Γ 2 ), we need −c 2 t 2 +c 3 = 0 and c 2 s 2 +c 1 = 0. Taking c 2 = 1 and c 0 = 0, we obtain that T commutes with the operator 10. In this example we consider self-adjoint bispectral Darboux transformations of the Bessel function ψ Be(ν) (x, y) of bidegree (2,2) corresponding to subspaces of the kernel of d 2 Be(ν),x .…”