## Integrable systems and extensions of the Calogero-Moser-Sutherland model

### Eric Dahlgren

Thursday, 13 December 2007, 10:30
Matfys Library

Abstract:
In this thesis the goal is to investigate the conjectured integrability of certain classical one--dimensional systems consisting of two types of particles interacting pairwise via an inverse square potential. Enabling us to study this assumption some basic concepts are introduced and explained in the first two chapters. Among these are the definition of classical integrability and the Lax representation of certain dynamical systems described by the Calogero-Moser-Sutherland model. Derived from known group integrals comes a parameter $\beta$ which in a natural way can be interpreted as a strength parameter in the potential of pairwise interacting particles. However, the hypothesized integrability of the systems in question is based on models in supersymmetric matrix spaces. Theses models give rise to not one but two strength parameters $\beta_1$ and $\beta_2$. Whenever $\beta_1\neq\beta_2$ these quantum systems can formally be connected to classical dynamical systems with two different kinds of particles. The hypothesis that these classical systems are integrable relies on the fact that for certain values of $\beta_1$ and $\beta_2$ exact wave functions to the corresponding quantum systems can be found.