INTRODUCTION The Calogero-Sutherland model has recently attracted some attention mainly because, in spite of its simplicity, it yields nontrivial results which contribute to shape our understanding of fractional statistics [1]. Its most remarkable property is that its ground state is given by a Jastrow wave-function which is the one dimensional analogue of the Laughlin wave-function. The excitations are described by quasiparticles which carry a fraction of the quantum numbers of the fundamental particles. The wave functions are simple enough to allow the computation of physical quantities such as the dynamical correlation functions. Recently, it has become clear that the spin version of this model is closely related to the known spin chains such as the XXX chain. The wave functions of the spin models can be obtained as simply as those of the Calogero-Sutherland model by diagonalising simple differential operators known as the Dunkl operators [2]. In these lectures, I restrict to the
We review recent results on new physical models constructed as PT-symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero-Moser-Sutherland type and nonlinear integrable field equations of Korteweg-de Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero-Moser-Sutherland models, we provide three alternative deformations: a complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real-valued field equations of nonlinear integrable systems; and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of Korteweg-de Vries type are studied with regard to different kinds of PT-symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.