Sutherland System Research Articles

Quantum versus classical integrability in Calogero$ndash$Moser systems

Calogero–Moser systems are classical and quantum integrable multiparticle dynamics defined for any root system Δ. The quantum Calogero systems having 1/q2 potential and a confining q2 potential and the Sutherland systems with 1/sin2q potentials have 'integer' energy spectra characterized by the root system Δ. Various quantities of the corresponding classical systems, e.g. minimum energy, frequencies of small oscillations, the eigenvalues of the classical Lax pair matrices etc, at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems. To our surprise, most of these classical data are also 'integers', or they appear to be 'quantized'. To be more precise, these quantities are polynomials of the coupling constant(s) with integer coefficients. The close relationship between quantum and classical integrability in Calogero–Moser systems deserves fuller analytical treatment, which would lead to better understanding of these systems and of integrable systems in general.

Factorized Weight Functions vs. Factorized Scattering

Starting from an extensive class of factorized weight functions W(p) on the N-dimensional torus ?, we construct an orthonormal base of symmetric N-variable polynomials for L 2 s (?,W(p)dp) via lexicographic ordering of the monomial symmetric functions (free boson states) and the Gram-Schmidt procedure. We show that the dominant asymptotics of these polynomials is factorized. As a corollary, we obtain a large class of quantum integrable soliton systems on the symmetric subspace of l 2(ℤ N ). The class of weight functions contains in particular the weight function yielding Macdonald polynomials. For that special case, the quantum soliton system can be viewed as the dual relativistic Calogero–Sutherland system.

Generalized Calogero–Moser–Sutherland models from geodesic motion on [formula omitted] group manifold

It is shown that geodesic motion on the GL(n, R) group manifold endowed with the bi-invariant metric ds 2= tr(g −1 dg) 2 corresponds to a generalization of the hyperbolic n-particle Calogero–Moser–Sutherland model. In particular, considering the motion on principal orbit stratum of the SO(n, R) group action, we arrive at dynamics of a generalized n-particle Calogero–Moser–Sutherland system with two types of internal degrees of freedom obeying SO(n, R)⊕SO(n, R) algebra. For the singular orbit strata of SO(n, R) group action the geodesic motion corresponds to certain deformations of the Calogero–Moser–Sutherland model in a sense of description of particles with different masses. The mass ratios depend on the type of singular orbit stratum and are determined by its degeneracy. Using reduction due to discrete and continuous symmetries of the system a relation to II A n Euler–Calogero–Moser–Sutherland model is demonstrated.

Super-Calogero–Moser–Sutherland systems and free super-oscillators: a mapping

We show that the supersymmetric rational Calogero-Moser-Sutherland (CMS) model of A_{N+1}-type is equivalent to a set of free super-oscillators, through a similarity transformation. We prescribe methods to construct the complete eigen-spectrum and the associated eigen-functions, both in supersymmetry preserving as well as supersymmetry breaking phases, from the free super-oscillator basis. Further we show that a wide class of super-Hamiltonians realizing dynamical OSp(2|2) supersymmetry, which also includes all types of rational super-CMS as a small subset, are equivalent to free super-oscillators. We study BC_{N+1}-type super-CMS model in some detail to understand the subtleties involved in this method.

Quantum integrable systems and Clebsch-Gordan series: II

The class of quantum integrable systems associated with root systems was introduced as a generalization of the Calogero - Sutherland systems. In this letter, a new property of such systems is proved to be valid. Namely, in the case of the potential , the series for the product of two wavefunctions coincides with the Clebsch - Gordan series. This gives the recursive relations for the wavefunctions of such systems and for generalized spherical functions related to them on symmetric spaces. One conjectures that the Clebsch - Gordan series is also unchanged under more general two-parametric deformation (-deformation).

Action-angle Maps and Scattering Theory for Some Finite-dimensional Integrable Systems. III. Sutherland Type Systems and their Duals

We present an explicit construction of an action-angle map for the nonrelativistic N - particle Sutherland system and for two different generalizations thereof, one of which may be viewed as a relativistic version. We use the map to obtain detailed information concerning dynamical issues such as oscillation periods and equilibria, and to obtain simple formulas for partition functions. The nonrelativistic and relativistic Sutherland systems give rise to dual integrable systems with a solitonic long-time asymptotics that is explicitly described. We show that the second generalization is self-dual, and that its reduced phase space can be densely embedded in \mathbf P^{N-1} with its standard Kähler form, yielding commuting global flows. In a certain limit the reduced action-angle map converges to the quotient of Fourier transformation on \mathbf C^N under the standard projection \mathbf C^N \backslash \{0\} → \mathbf P^{N-1} .

Quantum integrable systems of particles as gauge theories

We study quantum integrable systems of interacting particles from the point of view proposed by A. Gorsky and N. Nekrasov. We obtain the Sutherland system by a Hamiltonian reduction of an integrable system on the cotangent bundles to an affine su(N) algebra and show that it coincides with the Yang-Mills theory on a cylinder. We point out that there exists a tower of 2d quantum field theories. The top of this tower is the gauged G/C WZW model on a cylinder with an inserted Wilson line in an appropriate representation, which in our approach corresponds to Ruijsenaars` relativistic Calogero model. Its degeneration yields the 2d Yang-Mills theory, whose small radius limit is the Calogero model itself. We make some comments about the spectra and eigenstates of the models, which one can get from their equivalence with the field theories. Also we point out some possibilities of elliptic deformations of these constructions.

FRACTIONAL STATISTICS IN ONE DIMENSION: MODELING BY MEANS OF 1/x2 INTERACTION AND STATISTICAL MECHANICS

The equivalence of a quantum system of particles interacting with a two-body inverse square potential to a system of noninteracting particles obeying 1D fractional statistics (1D anyons), stated by Polychronakos for particles on a line, is studied for the cases where the interacting system is placed (i) into a harmonic potential on a line, and (ii) on a ring, with imposing periodic boundary conditions. In the first case, reducibility of the interacting system to the Calogero system is used to explore the statistical distribution for free 1D anyons. On a ring, the thermodynamic limit is discussed in terms of the thermodynamic (asymptotic) Bethe ansatz. Yang and Yang’s integral equation is treated in this case as describing the statistical mechanics of free 1D anyons. It gives a functional equation for the statistical distribution of 1D anyons consistent with the harmonic potential approach. We show that on a ring of a finite circumference, a system of two free 1D anyons is equivalent to a system of two particles interacting with an inverse sine square potential (the Sutherland system). We also discuss the relation to the statistical mechanics of free anyons in 2+1 dimensions.

Lattice integrable systems of Haldane-Shastry type.

We present a new lattice integrable system in one dimension of the Haldane-Shastry type. It consists of spins positioned at the static equilibrium positions of particles in a corresponding classical Calogero system and interacting through an exchange term with strength inversely proportional to the square of their distance. We achieve this by viewing the Haldane-Shastry system as a high-interaction limit of the Sutherland system of particles with internal degrees of freedom and identifying the same limit in a corresponding Calogero system. The commuting integrals of motion of this system are found using the exchange operator formalism.

Algebraic integrability for the Schr�dinger equation and finite reflection groups

Algebraic integrability of ann-dimensional Schrodinger equation means that it has more thann independent quantum integrals. Forn=1, the problem of describing such equations arose in the theory of finite-gap potentials. The present paper gives a construction which associates finite reflection groups (in particular, Weyl groups of simple Lie algebras) with algebraically integrable multidimensional Schrodinger equations. These equations correspond to special values of the parameters in the generalization of the Calogero—Sutherland system proposed by Olshanetsky and Perelomov. The analytic properties of a joint eigenfunction of the corresponding commutative rings of differential operators are described. Explicit expressions are obtained for the solution of the quantum Calogero—Sutherland problem for a special value of the coupling constant.

Tectonic Evolution and Paleogeography of Europe

The goal of this study was to use the tectonic framework of European craton to constrain our understanding of the sedimentary basins of Europe. An understanding of the amalgamation of the crustal blocks of Europe during the Caledonian, Hercynian, and Alpine orogenies was accomplished using an Evans and Sutherland system. Paleogeographic maps were ;made and integrated with the plate reconstruction with an eye toward how regional plate-scale events affect play elements in the basins. Europe is an artifact of Phanerozoic tectonic history, an amalgamation of crustal blocks without a precambrian nucleus of it own. This is in direct contrast of Africa, Asia, and North America. Multiple riftings and collisions created extremely complex mountain building during the Caledonian, Hercynian, Cimmerian, and Alpine orogenies. Basins are diverse, superimposed, and have long-lived tectonic histories with complex structuring and highly variable play elements. The Hercynian orogene set up the framework for northern European hydrocarbon systems. Its collapse set up the Apulian Mesozoic hydrocarbon system. Alpine deformation and tectonically related extension in turn set up the Neogene hydrocarbon systems of the Carpathians Pannonian basin and the Apennines. Eleven paleogeographic maps were completed at a scale of 1:5,000,000. There are four for the Paleozoic to showmore » the Hercynian orogeny and its subsequent foredeeps, and four for the Mesozoic, showing Tethyan rifting and associated subsidence, as well as the Cimmerian orogenies and start of Alpine deformation. The three time slices in the Cenozoic show the Alpine orogene and its foredeeps and the tectonically related extensional basins.« less

Classical Sutherland systems in the Morse potential

It is shown that the classical problem of motion of an arbitrary number of particles along a straight line with binary itneraction (shx)−2 placed under the action of the Morse potential can be solved by using the Lax representation. Exact solution of the equations of motion is reduced to calculation of the matrix exponentials. Existence conditions are determined for the points of equilibrium. The coordinates of particles at those points are defined by zeros of the generalized Laguerre polynomials, which allows us to find explicit expressions for the equilibrium-state energy and frequencies of small oscillations around equilibrium. Comparison is made with the results obtained earlier for the discrete spectrum of the corresponding quantum systems.

Quantum chaos of periodically pulsed systems: Underlying complete integrability.

The integrability underlying the quantum mechanics of noninte- grable pulsed systems is examined along the line exploited by Nakamura and Lakshmanan [Phys. Rev. Lett. 57, 1661 (1986)]. Coupled dynamical equations for both quasienergies and quasieigenfunctions (rather than matrix elements) with a nonintegrability parameter \ensuremath{\lambda} taken as ``time'' are shown to be reduced to a classical Sutherland's system with internal complex-vector space. Their complete integrability together with constants of motion are also exhibited.