On commuting matrix differential operators

Abstract:

If the differential expressions $P$ and $L$ are polynomials (over $\bb C$) of another differential expression they will obviously commute. To have a $P$ which does not arise in this way but satisfies $[P,L]=0$ is rare. Yet the question of when it happens has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs $(P,L)$. In this paper the question is answered in the case where the given expression $L$ has matrix-valued coefficients which are rational functions bounded at infinity or simply periodic functions bounded at the end of the period strip: if $Ly=zy$ has only meromorphic solutions then there exists a $P$ such that $[P,L]=0$ while $P$ and $L$ are not both polynomials of any other differential expression. The result is applied to the AKNS hierarchy where $L=JD+Q$ is a first order expression whose coefficients $J$ and $Q$ are $2\times 2$ matrices. It is therefore an elementary exercise to determine whether a given matrix $Q$ with rational or simply periodic coefficients is a stationary solution of an equation in the AKNS hierarchy.

Research supported in part by the US National Science Foundation under Grant No. DMS-9970299.

Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170, USA
rudi@math.uab.edu
http://www.math.uab.edu/rudi/