The Hamiltonian symmetry reduction of the geodesics system on a symmetric space of negative curvature by the maximal compact subgroup of the isometry group is investigated at an arbitrary value of the momentum map. Restricting to regular elements in the configuration space, the reduction generically yields a spin Calogero model with hyperbolic interaction potentials defined by the root system o...

It is shown that the Calogero-Moser models based on all root systems of the finite reflection groups (both the crystallographic and non-crystallographic cases) with the rational (with/without a harmonic confining potential), trigonometric and hyperbolic potentials can be simply supersymmetrised in terms of superpotentials. There is a universal formula for the supersymmetric ground state wavefun...

Noncommutative Ward’s conjecture is a noncommutative version of the original Ward’s conjecture which says that almost all integrable equations can be obtained from anti-selfdual Yang-Mills equations by reduction. In this paper, we prove that wide class of noncommutative integrable equations in both (2+1)and (1+1)-dimensions are actually reductions of noncommutative anti-self-dual Yang-Mills equ...

We study the bound states of the 1 + 1 dimensional Dirac equation with a scalar potential, which can also be interpreted as a position dependent “mass”, analytically as well as numerically. We derive a Prüfer-like representation for the Dirac equation, which can be used to derive a condition for the existence of bound states in terms of the fixed point of the nonlinear Prüfer equation for the a...

Calogero-Moser systems, which were originally discovered by specialists in integrable systems, are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representatio...

Calogero-Moser models and Toda models are well-known integrable multi-particle dynamical systems based on root systems associated with Lie algebras. The relation between these two types of integrable models is investigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairs of Calogero-Moser models are specified by t he representations of the reflection groups, which are not the ...

Liouville integrability of classical Calogero-Moser models is proved for models based on any root systems, including the non-crystallographic ones. It applies to all types of elliptic potentials, i.e. untwisted and twisted together with their degenerations (hyperbolic, trigonometric and rational), except for the rational potential models confined by a harmonic force. In this note we demonstrate...

We present a bridge between the KP soliton equations and the Calogero–Moser many-body systems through noncommutative algebraic geometry. The Calogero-Moser systems have a natural geometric interpretation as flows on spaces of spectral curves on a ruled surface. We explain how the meromorphic solutions of the KP hierarchy have an interpretation via a noncommutative ruled surface. Namely, we iden...