Ebook: Calogero—Moser— Sutherland Models

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.

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In the 1970s F. Calogero and D. Sutherland discovered that for certain potentials in one-dimensional systems, but for any number of particles, the Schr?dinger 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.

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In the 1970s F. Calogero and D. Sutherland discovered that for certain potentials in one-dimensional systems, but for any number of particles, the Schr?dinger 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.
Content:
Front Matter....Pages i-xxv
Classical Dynamical r-Matrices for Calogero—Moser Systems and Their Generalizations....Pages 1-21
Hidden Algebraic Structure of the Calogero—Sutherland Model, Integral Formula for Jack Polynomial and Their Relativistic Analog....Pages 23-35
Polynomial Eigenfunctions of the Calogero—Sutherland—Moser Models with Exchange Terms....Pages 37-51
The Theory of Lacunas and Quantum Integrable Systems....Pages 53-64
Canonical Forms for the C-Invariant Tensors....Pages 65-76
Tricks of the Trade: Relating and Deriving Solvable and Integrable Dynamical Systems....Pages 77-92
Classical and Quantum Partition Functions of the Calogero—Moser—Sutherland Model....Pages 93-116
The Meander Determinant and Its Generalizations....Pages 117-125
Differential Equations for Multivariable Hermite and Laguerre Polynomials....Pages 127-144
Heisenberg—Ising Spin Chain: Plancherel Decomposition and Chebyshev Polynomials....Pages 145-159
Ruijsenaars’s Commuting Difference System from Belavin’s Elliptic R-Matrix....Pages 161-176
Invariants and Eigenvectors for Quantum Heisenberg Chains with Elliptic Exchange....Pages 177-192
The Bispectral Involution as a Linearizing Map....Pages 193-202
On Some Quadratic Algebras: Jucys—Murphy and Dunkl Elements....Pages 203-220
Elliptic Solutions to Difference Nonlinear Equations and Nested Bethe Ansatz Equations....Pages 221-229
Creation Operators for the Calogero—Sutherland Model and Its Relativistic Version....Pages 231-248
New Exact Results for Quantum Impurity Problems....Pages 249-271
Painlev?—Calogero Correspondence....Pages 273-297
Yangian Symmetry in WZW Models....Pages 299-312
The Quantized Knizhnik—Zamolodchikov Equation in Tensor Products of Irreducible sl2-Modules....Pages 313-332
Gauge Fields and Interacting Particles....Pages 333-346
Generalizations of Calogero Systems....Pages 347-384
Three-Body Generalizations of the Sutherland Problem....Pages 385-398
On Relativistic Lam? Functions....Pages 399-410
Exact Solution for the Ground State of a One-Dimensional Quantum Lattice Gas with Coulomb—Like Interaction....Pages 411-420
The Distribution of the Largest Eigenvalue in the Gaussian Ensembles: ? = 1, 2, 4....Pages 421-440
Two-Body Elliptic Model in Proper Variables: Lie Algebraic Forms and Their Discretizations....Pages 441-449
Thermodynamics of Moser—Calogero Potentials and Seiberg—Witten Exact Solution....Pages 451-459
New Integrable Generalizations of the CMS Quantum Problem and Deformations of Root Systems....Pages 461-472
The Calogero Model: Integrable Structure and Orthogonal Basis....Pages 473-484
The Complex Calogero—Moser and KP Systems....Pages 485-495
Oscillator 9j-Symbols, Multdimensional Factorization Method, and Multivariable Krawtchouk Polynomials....Pages 497-506
....Pages 507-519