The Bispectral Involution as a Linearizing Map
In 1975, Moser introduced the map σ to linearize the motion of the Calogero—Moser—Sutherland model with rational inverse square potential. This linearizing map was unusual in that it turned out to be an involution on the phase space. In 1990, Wilson introduced an involution on a part of the Sato Grassmannian to demonstrate the bispectrality of certain Kadomtsev—Petviashvili (KP) solutions. Using the correspondence between particle systems and the poles of KP solutions, one may then also view Wilson’s bispectral involution as a map on the phase space of the particle system and compare them. As a result, we find that the bispectral involution is the linearizing map of the Calogero—Moser particle system and that it generates nonisospectral master symmetries of the KP flows. This chapter will review this material and then update it with reference to recent papers of Wilson, Shiota, Kasman—Rothstein, and Rothstein that simplify the proof and extend the results to include other integrable Hamiltonian systems of particle systems with linearizing involutions.
KeywordsParticle System Young Diagram Darboux Transformation Integrable Hamiltonian System Phase Space Variable
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