Algebraic Linearization of Dynamics of Calogero Type for any Coxeter Group

Calogero-Moser systems can be generalized for any root system (including the non-crystallographic cases). The algebraic linearization of the generalized Calogero-Moser systems and of their quadratic (resp. quartic) perturbations are discussed.Comment: LaTeX2e, 13 pages, no figure

Commuting Charges of the Quantum Korteweg-deVries and Boussinesq Theories from the Reduction of W(infinity) and W(1+infinity) Algebras

Integrability of the quantum Boussinesq equation for c=-2 is demonstrated by giving a recursive algorithm for generating explicit expressions for the infinite number of commuting charges based on a reduction of the W(infinity) algebra. These charges exist for all spins $s \geq 2$. Likewise, reductions of the W(infinity/2) and W((1+infinity)/2) algebras yield the commuting quantum charges for the quantum KdV equation at c=-2 and c=1/2, respectively.Comment: 11 pages, RevTe

Charged Vortex Dynamics in Ginzburg-Landau Theory of the Fractional Quantum Hall Effect

We write a Ginzburg-Landau Hamiltonian for a charged order parameter interacting with a background electromagnetic field in 2+1 dimensions. Using the method of Lund we derive a collective coordinate action for vortex defects in the order parameter and demonstrate that the vortices are charged. We examine the classical dynamics of the vortices and then quantize their motion, demonstrating that their peculiar classical motion is a result of the fact that the quantum motion takes place in the lowest Landau level. The classical and quantum motion in two dimensional regions with boundaries is also investigated. The quantum theory is not invariant under magnetic translations. Magnetic translations add total time derivative terms to the collective action, but no extra constants of the motion result.Comment: 28 pages + 1 Figure, new phyzzx macro (included), MAD/TH-92-0

Calogero-Moser Models III: Elliptic Potentials and Twisting

Universal Lax pairs of the root type with spectral parameter and independent coupling constants for twisted non-simply laced Calogero-Moser models are constructed. Together with the Lax pairs for the simply laced models and untwisted non-simply laced models presented in two previous papers, this completes the derivation of universal Lax pairs for all of the Calogero-Moser models based on root systems. As for the twisted models based on B_n, C_n and BC_nroot systems, a new type of potential term with independent coupling constants can be added without destroying integrability. They are called extended twisted models. All of the Lax pairs for the twisted models presented here are new, except for the one for the F_4 model based on the short roots. The Lax pairs for the twisted G_2 model have some novel features. Derivation of various functions, twisted and untwisted, appearing in the Lax pairs for elliptic potentials with the spectral parameter is provided. The origin of the spectral parameter is also naturally explained. The Lax pairs with spectral parameter, twisted and untwisted, for the hyperbolic, the trigonometric and the rational potential models are obtained as degenerate limits of those for the elliptic potential models.Comment: LaTeX2e with amsfonts.sty, 36 pages, no figure

Calogero-Moser Models V: Supersymmetry and Quantum Lax Pair

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 wavefunction. Since the bosonic part of each supersymmetric model is the usual quantum Calogero-Moser model, this gives a universal formula for its ground state wavefunction and energy, which is determined purely algebraically. Quantum Lax pair operators and conserved quantities for all the above Calogero-Moser models are established.Comment: LaTeX2e, 31 pages, no figure

Generalised Calogero-Moser models and universal Lax pair operators

Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H_3, H_4, and the dihedral group I_2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero-Moser models and for any choices of the potentials are constructed as linear combinations of the reflection operators. The consistency conditions are reduced to functional equations for the coefficient functions of the reflection operators in the Lax pair. There are only four types of such functional equations corresponding to the two-dimensional sub-root systems, A_2, B_2, G_2, and I_2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators, a powerful tool for solving quantum Calogero-Moser models.Comment: 37 pages, LaTeX2e, no macro, no figur

Quadratic Algebra associated with Rational Calogero-Moser Models

Classical Calogero-Moser models with rational potential are known to be superintegrable. That is, on top of the r involutive conserved quantities necessary for the integrability of a system with r degrees of freedom, they possess an additional set of r-1 algebraically and functionally independent globally defined conserved quantities. At the quantum level, Kuznetsov uncovered the existence of a quadratic algebra structure as an underlying key for superintegrability for the models based on A type root systems. Here we demonstrate in a universal way the quadratic algebra structure for quantum rational Calogero-Moser models based on any root systems.Comment: 19 pages, LaTeX2e, no figure

Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry

Inozemtsev models are classically integrable multi-particle dynamical systems related to Calogero-Moser models. Because of the additional q^6 (rational models) or sin^2(2q) (trigonometric models) potentials, their quantum versions are not exactly solvable in contrast with Calogero-Moser models. We show that quantum Inozemtsev models can be deformed to be a widest class of partly solvable (or quasi-exactly solvable) multi-particle dynamical systems. They posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A new method for identifying and solving quasi-exactly solvable systems, the method of pre-superpotential, is presented.Comment: LaTeX2e 28 pages, no figure

The Crystal Structure of a Proposed Hinesol Intermediate

The structure of a proposed intermediate of the naturally occurring sesquiterpene hinesol was determined by X-ray analysis. The structure was solved using the heavy-atom technique and refined to an R index of 0-065 for 1739 non-zero reflections. The results of the analysis revealed that the intermediate had a structure incompatible with conversion to hinesol