自旋Calogero粒子与矩阵KP族的双谱解
Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy
摘要：
利用交换子与单位元相差秩矩阵的$n\times n$矩阵对构造双谱微分算子，其$r\times r$矩阵系数满足矩阵KP族的Lax方程.此外，这些算子的双谱对合对自旋Calogero粒子系统的相空间，这些对所代表的动力学意义。在$r=1$的情况下，这再现了Wilson和其他人在20世纪90年代将（无自旋）Calogero-Moser系统与（标量）微分算子的双谱性联系起来的著名结果。这类新的双谱矩阵微分算子对$(L, Λ)$与以前研究的不同之处在于$L$从左边起作用，而$Λ$从右边起作用于一个共同的$r\times r$特征矩阵。
英文摘要：
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 $(L, Λ)$ 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 $r\times r$ eigenmatrix.
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