Bispectrality of $N$-Component KP Wave Functions: A Study in Non-Commutativity

Abstract

A wave function of the $N$-component KP Hierarchy with continuous flows determined by an invertible matrix $H$ is constructed from the choice of an $MN$-dimensional space of finitely-supported vector distributions. This wave function is shown to be an eigenfunction for a ring of matrix differential operators in $x$ having eigenvalues that are matrix functions of the spectral parameter $z$. If the space of distributions is invariant under left multiplication by $H$, then a matrix coefficient differential-translation operator in $z$ is shown to share this eigenfunction and have an eigenvalue that is a matrix function of $x$. This paper not only generates new examples of bispectral operators, it also explores the consequences of non-commutativity for techniques and objects used in previous investigations.