Spectral Difference Equations Satisfied by Kp Soliton Wavefunctions

It is by now well known that the wave functions of rational solutions to the KP hierarchy which can be achieved as limits of the pure n-soliton solutions satisfy an eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as “bispectrality” and has proved to be both interesting and useful. In this note, it is shown that all pure soliton solutions of the KP hierarchy (as well as their rational degenerations) satisfy an eigenvalue equation for a non-local operator constructed by composing ordinary differential operators in the spectral parameter with translation operators in the spectral parameter, and therefore have a form of bispectrality as well.