Bispectral KP solutions and linearization of Calogero-Moser particle systems

Rational and soliton solutions of the KP hierarchy in the subgrassmannianGr₁ are studied within the context of finite dimensional dual grassmannians. In the rational case, properties of the tau function, , which are equivalent to bispectrality of the associated wave function, , are identified. In particular, it is shown that there exists a bound on the degree of all time, variables in if and only if is a rank one bispectral wave function. The action of the bispectral involution, , in the generic rational case is determined explicitly in terms of dual grassmannian parameters. Using the correspondence between rational solutions, and particle systems, it is demonstrated that is a linearizing map of the Calogero-Moser particle system and is essentially the map introduced by Airault, McKean and Moser in 1977 [2].Research supported by NSA Grant MDA904-92-H-3032