## 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.