The Calogero (Moser(Sutherland)) system is an autonomous integrable  Hamiltonian   system of n particles on the line interacting with inverse square potential (or Weierstrass-p function). The non-interacting part is a classically integrable Hamiltonian (e.g. the harmonic oscillator).

The principal goal of the talk is to explain how the integrability survives if we replace the single-particle Hamiltonian by any of the Hamiltonian for the six Painleve' equations, hence turning the system into a time-dependent, interacting dynamics.

In this case, as we would expect, the isospectral character of the associated Lax system, is replaced by an isomonodromic evolution that linearizes non-commutative  versions (i.e. matrix-valued) of the six Painleve' equations. The result solves  a conjecture posed by T. Takasaki i 2010.  Joint work with M. Cafasso and V. Rubtsov.

 

Notes: this is an event within the "informal lunch seminar" series of the ISMP group

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