Reflective prolate-spheroidal operators and the KP/KdV equations.

Commuting integral and differential operators connect the topics of signal processing, random matrix theory, and integrable systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the Korteweg-de Vries (KdV) equation. We prove a general theorem that the integral operator associated to every wave function in the infinite-dimensional adelic Grassmannian [Formula: see text] of Wilson always reflects a differential operator (in the sense of below).