Calogero-Moser-Sutherland Models

by Van Diejen, Jan Felipe; Vinet, Luc

• ISBN13: 9780387989686
• ISBN10: 0387989684
• Format: Hardcover
• Copyright: 2000-05-01
• Publisher: Springer Verlag

Summary

In the 1970s F. Calogero and D. Sutherland discovered that for certain potentials in one-dimensional systems, but for any number of particles, the Schrödinger eigenvalue problem is exactly solvable. Until then, there was only one known nontrivial example of an exactly solvable quantum multi-particle problem. J. Moser subsequently showed that the classical counterparts to these models is also amenable to an exact analytical approach. The last decade has witnessed a true explosion of activities involving Calogero-Moser-Sutherland models, and these now play a role in research areas ranging from theoretical physics (such as soliton theory, quantum field theory, string theory, solvable models of statistical mechanics, condensed matter physics, and quantum chaos) to pure mathematics (such as representation theory, harmonic analysis, theory of special functions, combinatorics of symmetric functions, dynamical systems, random matrix theory, and complex geometry). The aim of this volume is to provide an overview of the many branches into which research on CMS systems has diversified in recent years. The contributions are by leading researchers from various disciplines in whose work CMS systems appear, either as the topic of investigation itself or as a tool for further applications.

Table of Contents

Series Preface	v
Preface	vii
Contributors	xxi
Classical Dynamical r-Matrices for Calogero--Moser Systems and Their Generalizations	1	(22)
J. Avan
Introduction	1	(1)
Preliminaries	2	(2)
Liouville Theorem	2	(1)
Lax Pair Formulation	3	(1)
The r-Matrix Structure	3	(1)
The Classical Yang-Baxter Equation	3	(1)
Hamiltonian Reduction and r-Matrices	4	(8)
General Hamiltonian Reduction	4	(2)
The Case N = T*G	6	(1)
The Calogero-Moser Models	7	(3)
Two Examples	10	(2)
The Dynamical r-Matrices of Calogero and Ruijsenaars Models	12	(5)
References	17	(6)
Hidden Algebraic Structure of the Calogero-Sutherland Model, Integral Formula for Jack Polynomial and Their Relativistic Analog	23	(14)
Hidetoshi Awata
Introduction	23	(1)
Calogero-Sutherland Model and Jack Polynomial	24	(2)
Integral Formula for Jack Polynomial	26	(2)
Relation with Virasoro Singular Vectors	28	(2)
Macdonald Polynomial and q-Virasoro Algebra	30	(2)
Boson Realization for q-Virasoro Algebra and Level One Elliptic Affine Lie Algebra	32	(2)
References	34	(3)
Polynomial Eigenfunctions of the Calogero-Sutherland-Moser Models with Exchange Terms	37	(16)
T. H. Baker
C. F. Dunkl
P. J. Forrester
Introduction	37	(4)
The Periodic Model	37	(3)
The Linear Model	40	(1)
Eigenfunctions of Prescribed Symmetry	41	(5)
Expansions in Terms of Eη	41	(4)
Normalization	45	(1)
q-Analogs	46	(4)
References	50	(3)
The Theory of Lacunas and Quantum Integrable Systems	53	(12)
Yuri Yu. Berest
Introduction	53	(1)
Hyperbolic Operators on Root Systems	54	(4)
Hhperbolic Polynomials and Associated Riesz Integrals	54	(2)
Invariant Hyperbolic Operators	56	(1)
The Representation Theorem	57	(1)
Lacunas and Topology of Algebraic Surfaces	58	(5)
Vector Fields and Cycles	58	(2)
The Herglotz--Petrovsky--Leray Formulas	60	(2)
Products of Wave Operators	62	(1)
References	63	(2)
Canonical Forms for the C-Invariant Tensors	65	(12)
Oleg I. Bogoyavlenskij
Introduction	65	(3)
C-Invariant Differential 1-Forms	68	(2)
General C-Invariant (0,2) Tensors	70	(1)
Characteristic Polynomial of Any C-Invariant (1,1) Tensor Is a Perfect Square	71	(2)
Nijenhuis Tensor and C-Invariant (1,2) Tensors	73	(2)
References	75	(2)
R-Matrices, Generalized Inverses, and Calogero-Moser--Sutherland Models	77	(16)
H. W. Braden
Introduction	77	(3)
R-Matrices	80	(2)
Four Results	82	(3)
Application	85	(4)
Discussion	89	(1)
References	90	(3)
Tricks of the Trade: Relating and Deriving Solvable and Integrable Dynamical Systems	93	(24)
Francesco Calogero
Introduction	93	(1)
Survey of Solvable and Integrable Many-Body Models	94	(4)
Tricks	98	(16)
Time-Dependent Rescalings of Dependent Variables and of Time	98	(2)
From One to More Kinds of Particles by Shifting the Dependent Variables	100	(1)
``Duplications'': Configurations that are Preserved Throughout the Motion	101	(6)
How to Get Rid of Velocity-Dependent Forces	107	(1)
From Two-Body to Nearest-Neighbor Forces	108	(1)
Infinite Rescalings, and the Use of Special Solutions, to Identify Solvable Nonautonomous Models	109	(2)
Two-Dimensional Models via Complexification	111	(3)
Envoi	114	(1)
References	115	(2)
Classical and Quantum Partition Functions of the Calogero--Moser--Sutherland Model	117	(10)
Ph. Choquard
Preamble and Summary	117	(1)
The CSM Model	118	(1)
Thermodynamics: Classical Case	119	(2)
Thermodynamics: Quantum Case	121	(3)
References	124	(3)
The Meander Determinant and Its Generalizations	127	(18)
P. Di Francesco
Introduction	127	(1)
Meanders: Definitions and Reformulations	128	(5)
Definitions	128	(1)
Temperley--Lieb Algebra	129	(2)
Meander Determinant	131	(2)
Road/River Generalizations	133	(3)
Semimeanders	133	(1)
Crossing Meanders and Brauer Algebra	134	(2)
SU (N) Generalizations	136	(7)
Hecke Algebra and Its Quotients	136	(1)
SU (N) Meanders	137	(3)
SU (N) Meander Determinants	140	(3)
References	143	(2)
Differential Equations for Multivariable Hermite and Languerre Polynomials	145	(16)
J. F. van Diejen
Introduction	145	(1)
Multivariable Hermite and Laguerre Polynomials	146	(1)
Differential Equations	147	(5)
Proof of the Differential Equations	152	(5)
Relation to the Quantum Calogero Model	157	(1)
References	158	(3)
Quantum Currents Realization of the Elliptic Quantum Groups Eτ,η(sl2)	161	(16)
B. Enriquez
Elliptic Quantum Groups	162	(2)
Function Spaces Associated with Elliptic Curves	162	(1)
Elliptic R--Matrix	163	(1)
The Dynamical Yang--Baxter Equation	163	(1)
Elliptic Quantum Groups	163	(1)
Another Presentation	164	(1)
Quantum Currents Algebra	164	(4)
Classical Structures	165	(1)
Quantum Algebra	166	(1)
Coproducts	167	(1)
The Realization	168	(2)
Half-Currents	168	(1)
Realization	169	(1)
About the Proof	170	(5)
A Rational Analog	170	(1)
Subalgebras of Hol(C \ Γ, Uhg(τ)⊗2) and Hol (C \ Γ, Uhg(τ)&otimes3)	171	(1)
Decomposition of F	172	(1)
Twisted Cocycle Equation	173	(1)
The BBB Result	174	(1)
End of the Proof	174	(1)
References	175	(2)
Heisenberg--Ising Spin Chain: Plancherel Decomposition and Chebyshev Polynomials	177	(16)
Eugene Gutkin
The XXZ Hamiltonian and the Intertwining Operators	179	(2)
Bethe Ansatz Eigenstates	181	(1)
Chebyshev Circles and Plancherel Sets	182	(3)
The Plancherel Formula	185	(1)
Sketch of the Proof of Plancherel Decomposition	186	(2)
Induction on the Magnon Number	186	(1)
Hyperbolic Region of the Anisotropy Parameter	187	(1)
Elliptic Region of the Anisotropy Parameter	187	(1)
Examples	188	(2)
2-Magnon Sector	188	(1)
3-Magnon Sector	189	(1)
4-Magnon Sector	189	(1)
References	190	(3)
Ruijsenaar's Commuting Difference System from Belavin's Elliptic R-Matrix	193	(10)
Koji Hasegawa
Introduction	193	(1)
The Difference Operators	194	(5)
An Invariant Subspace Spanned by Symmetric Theta Functions	199	(1)
References	200	(3)
Invariants and Eigenvectors for Quantum Heisenberg Chains with Elliptic Exchange	203	(18)
V. I. Inozemtsev
Introduction	203	(1)
Notation and Statements	204	(3)
Proofs	207	(4)
Proofs of Auxiliary Statements	207	(1)
Proof of Proposition 1	208	(3)
Proof of Proposition 2	211	(1)
The Eigenvectors	211	(8)
References	219	(2)
The Bispectral Involution as a Linearizing Map	221	(10)
Alex Kasman
Moser's Linearizing Involution	221	(2)
The Bispectral Involution	223	(2)
Bispectrality of Ordinary Differential Operators	223	(1)
Brief Review of Sato Grassmannian	223	(1)
The Bispectral Involution and Grad	224	(1)
Dynamics: the Bispectral Flow	224	(1)
Relating the Two Involutions	225	(1)
Tau Functions	225	(1)
Conclusions	226	(1)
References	227	(4)
On Some Quadratic Algebras: Jucys--Murphy and Dunkl Elements	231	(18)
Anatol N. Kirillov
Introduction	231	(2)
Quadratic Algebra Bn	233	(3)
Jucys--Murphy Elements	236	(3)
Quadratic Algebra Gn	239	(1)
Dunkl Elements	240	(1)
Dunkl and Jucys-Murphy Elements	241	(5)
References	246	(3)
Elliptic Solutions to Difference Nonlinear Equations and Nested Bethe Ansatz Equations	249	(24)
I. Krichever
Introduction	249	(2)
Generating Problem	251	(12)
Hamiltonian Theory of the CM-Type Systems	263	(6)
References	269	(4)
Creation Operators for the Calogero--Sutherland Model and Its Relativistic Version	273	(26)
Luc Lapointe
Luc Vinet
Introduction	273	(1)
Notations and Definitions	274	(2)
Jack Polynomials and the Calogero--Sutherland Model	276	(2)
Goal and Outline	278	(1)
Dunkl Operators	279	(2)
Rodrigues Formula	281	(7)
Additional Results	288	(2)
Another Set of Creation Operators	290	(3)
Relativistic Generalization: The Macdonald Polynomials	293	(3)
References	296	(3)
New Exact Results for Quantum Impurity Problems	299	(14)
F. Lesage
H. Saleur
P. Simonetti
Introduction	299	(1)
Connection with Kondo Models and 1D Wires	300	(3)
Perturbation Theory and Jack Polynomials	303	(1)
Integrable Basis	304	(4)
Transport Properties for the 1D Wires	308	(2)
Conclusion	310	(1)
References	311	(2)
Painleve--Calogero Correspondence	313	(20)
A. M. Levin
M. A. Olshanetsky
Introduction	313	(4)
Painleve VI and Calogero Equations	314	(3)
Symplectic Reduction	317	(5)
Upstairs Extended Phase Space	317	(1)
Symmetries	318	(1)
Symplectic Reduction with Respect to G1	318	(2)
Factorization with Respect to the Diffeomorphisms G0	320	(1)
The Hierarchies of the Isomonodromic Deformations	321	(1)
Relations to the Hitchin Systems and the KZB Equations	322	(2)
Scaling Limit	322	(1)
About KZB	323	(1)
Genus Zero---Schlesinger's Equation	324	(1)
Genus One---Elliptic Schlesinger, Painleve VI, Etc	325	(5)
Deformations of Elliptic Curves	325	(1)
Flat Bundles on a Family of Elliptic Curves	326	(1)
Symmetries	327	(1)
Symplectic Form	327	(3)
References	330	(3)
Yangian Symmetry in WZW Models	333	(14)
Z. Maassarani
P. Mathieu
Conformal Field Theory: The ``Field Description''	333	(1)
Integrable Quantum Field Theory: The ``Particle Description''	334	(1)
The Yangian Symmetry of the su (2)1-WZW Model	335	(2)
Nonlocal Charges and S-Matrices	337	(2)
Nonlocal Currents in WZW Models	339	(2)
Nonlocal Charges in WZW Models and Comultiplication	341	(1)
Yangian Relations	342	(1)
Yangians Densities as Logarithmic Operators	342	(2)
Conclusions	344	(1)
References	344	(3)
The Quantized Knizhnik--Zamolodchikov Equation in Tensor Products of Irreducible Sl2-Modules	347	(38)
E. Mukhin
A. Varchenko
Introduction	347	(2)
General Definitions and Notations	349	(5)
The Lie Algebra Sl2	349	(1)
The Algebra Uq Sl2	350	(1)
The Rational R-Matrix	351	(1)
The Trigonometric R-Matrix	352	(1)
The qKZ Equation	353	(1)
The Hypergeometric Pairing	354	(6)
The Phase Function	354	(1)
Actions of the Symmetric Group	354	(1)
Rational Weight Functions	355	(1)
Trigonometric Weight Functions	355	(3)
Hypergeometric Integrals	358	(2)
Main Results	360	(9)
Analytic Continuation. The Case Im μ ≠ 0	360	(2)
Analytic Continuation	362	(1)
Solutions of qKZ with Values in Irreducible Representations	363	(1)
Determinant of the Hypergeometric Pairing	364	(3)
Asymptotic Solutions	367	(2)
Proofs	369	(14)
Proof of Theorem 7	370	(8)
Proof of Theorems 8 and 9	378	(3)
Proof of Theorem 12	381	(1)
Proof of Theorem 1	382	(1)
References	383	(2)
Gauge Fields and Interacting Particles	385	(14)
N. Nekrasov
Introduction	385	(1)
The Cast	386	(2)
The Hamiltonian Reduction and Gauge Theories	388	(4)
Supersymmetric Gauge Theories in Four Dimensions	392	(2)
Stringy Derivation	394	(1)
Conclusions and Speculations	395	(1)
References	396	(3)
Generalizations of Calogero Systems	399	(12)
Alexios P. Polychronakos
References	408	(3)
Three-Body Generalizations of the Sutherland Problem	411	(10)
C. Quesne
Introduction	411	(1)
Exact Solvability of the Pure Three-Body Problem	412	(3)
Two-and Three-body Problem with Internal Degrees of Freedom	415	(4)
References	419	(2)
On Relativistic Lame Functions	421	(20)
S. N. M. Ruijsenaars
Introduction	421	(3)
The Nonrelativistic Integer g Case	424	(3)
The Relativistic Integer g Case	427	(4)
Eigenfunctions for a Dense Parameter Set	431	(4)
Concluding Remarks	435	(3)
References	438	(3)
Exact Solution for the Ground State of a One-Dimensional Quantum Lattice Gas with Coulomb--Like Interaction	441	(10)
Bill Sutherland
Introduction	441	(1)
Summary of Paper I	442	(2)
Summary of Paper II	444	(1)
A Lattice Version	445	(1)
Ground-State Correlations	446	(2)
References	448	(3)
Differential Operators that Commute with the r--2-type Hamiltonian	451	(10)
Kenji Taniguchi
Introduction	451	(1)
Commutative Families of Weyl Group Invariant Differential Operators	452	(1)
Periodic Potential Cases	453	(3)
Rational Potential Cases	456	(3)
References	459	(2)
The Distribution of the Largest Eigenvalue in the Gaussian Ensembles: β = 1, 2, 4	461	(12)
Craig A. Tracy
Harold Widom
Introduction	461	(1)
Why β=2 is the Simplest Case	462	(1)
Edge Scaling Limit	463	(2)
Cases β=1 and 4	465	(1)
Idea of Proof and Results	466	(2)
How ``Useful'' Are These Scaling Functions?	468	(2)
Questions of Universality	470	(1)
References	471	(2)
Two-Body Elliptic Model in Proper Variables: Lie Algebraic Forms and Their Discretizations	473	(12)
Alexander Turbiner
Introduction	473	(1)
Lie Algebraic Analysis	473	(4)
Translation-Invariant Discretization	477	(3)
Dilatation-Invariant Discretization	480	(2)
References	482	(3)
Yangian Gelfand--Zetlin Bases, glN-Jack Polynomials, and Computation of Dynamical Correlation Functions in the Spin Calogero--Sutherland Model	485	(12)
Denis Uglov
Introduction	485	(3)
Cherednik--Dunkl operators, Yangian, Yangian Gelfand--Zetlin Bases	488	(1)
The Wedge Basis and the Scalar Product	489	(1)
Spin Calogero--Sutherland Model in the Framework of Symmetric Polynomials	490	(1)
Dynamical Spin-Density and Density Correlation Function for N = 2	491	(2)
References	493	(4)
Thermodynamics of Moser--Calogero Potentials and Seiberg--Witten Exact Solution	497	(10)
K. L. Vaninsky
Introduction	497	(1)
Gibbs' States for NLS in Action-Angle Variables	498	(1)
Thermodynamics of the Moser--Calogero Potentials	498	(1)
Sketch of the Computations	499	(3)
Trace Formula for Particles with Elliptic Potential	502	(2)
Symplectic Structures and Volume Elements for NLS	504	(1)
References	505	(2)
New Integrable Generalizations of the CMS Quantum Problem and Deformations of Root Systems	507	(14)
A. P. Veselov
Basic Construction and Coxeter Root Systems	509	(2)
Deformations of the Root Systems and New Integrable Quantum Problems	511	(6)
Concluding Remarks	517	(1)
References	518	(3)
The Calogero Model: Integrable Structure and Orthogonal Basis	521	(18)
Miki Wadati
Hideaki Ujino
Introduction	521	(2)
Integrability and Algebraic Structure	523	(2)
Perelomov Basis	525	(1)
Diagonalization of I2	526	(2)
Hi-Jack Polynomials	528	(7)
References	535	(4)
The Complex Calogero--Moser and KP Systems	539	(10)
George Wilson
Introduction	539	(1)
The Completed Phase Space for the Complex Calogero--Moser Flows	540	(1)
Geometry of Rational KP Solutions	541	(4)
Applications	545	(1)
4.1	545	(1)
4.2	545	(1)
The Spin Generalization	545	(1)
References	546	(3)
Oscillator 9j-Symbols, Multdimensional Factorization Method, and Multivariable Krawtchouk Polynomials	549
Alexei Zhedanov
Introduction	549
Oscillator Algebra, Its Addition Rule, and 9j-Symbols	550
General Scheme of Multidimensional Factorization Method	553
Algebraic Ansatz and a Special Solution of the Factorized Chain	555
Difference Operators and Multivariable Polynomials	557
Conclusion	559
References	560