R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators

We shall give a certain trigonometric R-matrix associated with each root system by using affine Hecke algebras. From this R-matrix, we derive a quantum Knizhnik-Zamolodchikov equation after Cherednik, and show that the solutions of this KZ equation yield eigenfunctions of Macdonald's difference operators.     

A Class of Integrable Spin Calogero-Moser Systems

 We introduce a class of spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models (called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and gauge transformations. For Lie algebras of A _n -type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided by a general framework which we develop here using dynamical Lie algebroids. Received: 19 October 2001 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research partially supported by NSF grant DMS00-72171.     

Calogero-Moser and Toda systems for twisted and untwisted affine Lie algebras

The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a general simple Lie algebra G are shown to scale to the (affine) Toda Hamiltonian and Lax pair. The limit consists in taking the elliptic modulus τ and the Calogero-Moser couplings m to infinity, while keeping fixed the combination M = m e^iδθτ for some exponent δ. Critical scaling limits arise when 1/δ equals the Coxeter number or the dual Coxeter number for the untwisted and twisted Calogero-Moser systems respectively; the limit consists then of the Toda system for the affine Lie algebras G⁽¹⁾ and (G⁽¹⁾)^V. The limits of the untwisted or twisted Calogero-Moser system, for δ less than these critical values, but non-zero, consists of the ordinary Toda system, while for δ = 0, it consists of the trigonometric Calogero-Moser systems for the algebras G and G^V respectively.     

On the solvability of the Calogero-Moser system in some external potentials and of other related hamiltonian systems

Explicit solutions for the Calogero-Moser system in a large class of external potentials including also some quartic oscillators are derived. The method applied extends to many other nonnatural hamiltonians related with the Calogero-Moser system.     

三角Calogero-Moser模型的非动力学r矩阵结构

通过一定规范变换,构造了三角Calogero–Moser模型一种新的Lax算子,使其具有相应的非动力学r矩阵结构.同时发现该r矩阵结构与三角Ruijsenaars–Schneider模型的r矩阵完全相同.     

Separation of Variables for the Ruijsenaars System

We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variables is verified with the use of the r-matrix technique. The explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n=2 and n=3. Taking the nonrelativistic limit we also construct a separation of variables for the elliptic Calogero-Moser system. Received: 10 January 1997 / Accepted: 1 April 1997     

On the integrability of the Calogero-Moser system in an external quartic potential and other many-body systems

The complete integrability of the Calogero-Moser system in an external quartic potential is proved and the Bäcklund transformation for this system is found. The admissible, within the isospectral deformation method, external potentials for other many-body systems are discussed.     

Classical integrable systems and gauge field theories

In this review we consider the Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimensions. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider the Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL (N, C) and explain the Symplectic Hecke correspondence for these systems. The text was submitted by the author in English.     

Quantum vs. Classical Integrability in Ruijsenaars–Schneider and Calogero–Moser Systems

The relationship (resemblance and/or contrast) between quantum and classical integrability in Ruijsenaars-Schneider (R-S) and Calogero-Moser systems is addressed. The classical Calogero and Sutherland systems (based on any root system) at equilibrium have many remarkable properties; for example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all integer valued. These are related to the energy eigenvalues of the quantum Calogero and Sutherland systems. Similar features and results hold for the R-S type of integrable systems based on the classical root systems.     

THE LAX PAIR FOR C2-TYPE RUIJSENAARS-SCHNEIDER MODEL

We study the C₂ Ruijsenaars-Schneider model with interaction potential of trigonometric type. The Lax pairs for the model with and without spectral parameters are constructed. Also given are the involutive Hamiltonians for the system. Taking a non-relativistic limit, we obtain the Lax pair of C₂ Calogero-Moser model.     

A generalisation of the Calogero-Moser system

A generalised Calogero-Moser many-body system, in which the particles possess extra internal degrees of freedom, is introduced and solved by the method of Olshanetsky and Perelomov.Further, the hierarchy of flows commuting with this system is obtained from a hierarchy of linear systems possessing SL(N) symmetry, by the usual method of reduction; thus they are shown to be integrable.     

2D LOCUS CONFIGURATIONS AND THE TRIGONOMETRIC CALOGERO–MOSER SYSTEM

A central hyperplane arrangement in ?² with multiplicity is called a "locus configuration" if it satisfies a series of "locus equations" on each hyperplane. Following [4], we demonstrate that the first locus equation for each hyperplane corresponds to a force-balancing equation on a related interacting particle system on ?*: the charged trigonometric Calogero-Moser system. When the particles lie on S¹ ? ?*, there is a unique equilibrium for this system. For certain classes of particle weight, this is enough to show that all the locus equations are satisfied, producing explicit examples of real locus configurations. This in turn produces new examples of Schrödinger operators with Baker–Akhiezer functions.     

20 MeV以下n+6Li反应的理论计算

核反应R矩阵理论是研究轻核反应以及中重和重核共振能区核反应的重要理论方法。包含能级矩阵非对角元贡献的完全R矩阵理论在理论上比较严格。根据不同的假定和近似得到不同的R矩阵计算方法。新编的R矩阵程序FDRR包含了4种计算方法,包括约化道多能级Breit-Wigner 方法、完全约化R矩阵方法、非对角化能级位移约化R矩阵方法及对角化能级位移约化R矩阵方法。可计算轻核各种两体反应道的截面、角分布。利用FDRR程序对n+6Li 反应20 MeV以下能区进行理论分析和计算,理论计算结果与实验数据进行了比较分析,理论计算得到的截面和角分布与实验数据符合得很好。R-matrix theory is an important theory of light, medium and heavy mass nuclide nuclear reaction in the resonance energy range. Full R-matrix formalism contains the un-diagonal element of energy levels matrix and it is rigorous in theory. Because of different assumptions and approximations, many kinds of R-matrix methods are obtained. The new R-matrix code FDRR is presented and includes 4 kinds of R-matrix methods, reduced multi-level Breit-Wigner R-matrix method, full reduced R-matrix method, un-diagonal energy shift reduced Rmatrixn method, and diagonal energy shift reduced R-matrix method. It can be used for calculating integral cross sections and angular distributions of 2-bodies reactions. The cross sections and angular distributions of n+6Li reaction are calculated and analyzed by FDRR code below 20 MeV. The calculation results are compared with the experimental data, and the cross sections and angular distributions are in good agreement with experimental data.     

Calogero-Moser Lax pairs with spectral parameter for general Lie algebras

We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite-dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable systems, naturally associated with twisted affine Lie algebras, and construct their Lax operators with spectral parameter (except in the case of G₂).     

Vector Bundles and Lax Equations on Algebraic Curves

The Hamiltonian theory of zero-curvature equations with spectral parameter on an arbitrary compact Riemann surface is constructed. It is shown that the equations can be seen as commuting flows of an infinite-dimensional field generalization of the Hitchin system. The field analog of the elliptic Calogero-Moser system is proposed. An explicit parameterization of Hitchin system based on the Tyurin parameters for stable holomorphic vector bundles on algebraic curves is obtained. Received: 25 September 2001 / Accepted: 22 December 2001     

Liouville Integrability of a Class of Integrable Spin Calogero-Moser Systems and Exponents of Simple Lie Algebras

In previous work, we introduced a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here the main purpose is to establish the Liouville integrability of these systems by a uniform method based on evaluating the primitive invariants of Chevalley on the Lax operators with spectral parameter. As part of our analysis, we will develop several results concerning the algebra of invariant polynomials on simple Lie algebras and their expansions.     

Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

Pairs of n×n matrices whose commutator differ from the identity by a matrix of rank r are used to construct bispectral differential operators with r×r matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case r = 1, this reproduces well-known results of Wilson and others from the 1990’s relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators.      

The propagator of the Calogero-Moser system in an external quadratic potential

We obtain the Hamilton operator of the Calogero-Moser quantum system in an external quadratic potential by conjugating the radial part for the action of SO(n) by conjugacy of the Hamilton operator of the quantum harmonic oscillator on the Euclidean vector space of real symmetric matrices. Then, with Mehler's formula, we derive the propagator of the problem. We also investigate some schemes to change the interaction constant. For two-particle systems, we obtain explicit formulae, whereas for many-particle systems, we reduce the computation of the propagator to finding a definite integral. We give also the short time approximation, the energy levels and the trace of the propagation operator.     

Quantum Dynamical R-Matrices and Quantum Frobenius Group

We propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parametrization of the cotangent bundle over . In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical -matrix. Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system. Using the dual parametrization of the cotangent bundle we also derive the algebra for the L-operator of the hyperbolic Calogero-Moser system. Received: 24 January 1997 / Accepted: 17 March 1997     

低能电子被水分子散射总截面的计算

基于固定核近似原理,利用R矩阵方法和分子的R矩阵程序计算了在4-20 eV能量范围内电子被水分子弹性散射的总碰撞截面,同时讨论了R矩阵半径的取值对总截面的影响.并将结果与实验值及其它理论计算作了比较,结果显示这种计算方法对于电子分子碰撞是很有效的.