Quantum integrable systems related to lie algebras

doi:10.1016/0370-1573(83)90018-2

Physics Reports

Volume 94, Issue 6, March 1983, Pages 313-404

https://doi.org/10.1016/0370-1573(83)90018-2 Get rights and content

Abstract

Some quantum integrable finite-dimensional systems related to Lie algebras are considered. This review continues the previous review of the same authors [83] devoted to the classical aspects of these systems. The dynamics of some of these systems is closely related to free motion in symmetric spaces. Using this connection with the theory of symmetric spaces some results such as the forms of spectra, wave functions, S-matrices, quantum integrals of motion are derived. In specific cases the considered systems describe the one-dimensional n-body systems interacting pairwise via potentials g² v(q) of the following 5 types: v_I(q) = q⁻², v_II(q) = sinh⁻² q, v_III(q) = sin⁻² q, $v_{IV} (q) = P (q)$ , v_V(q) = q⁻² + ω²q². Here $P (q)$ is the Weierstrass function, so that the first three cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbour potential exp(q_jq_{j+ 1}) is moreover considered.

This review presents from a general and universal point of view results obtained mainly over the past fifteen years. Besides, it contains some new results both of physical and mathematical interest.

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View full text

Quantum integrable systems related to lie algebras

Abstract

Ann. Phys.

J. Math. Anal. Appl.

Lett. Math. Phys.

Phys. Rev.

J. Math. Phys.

Ann. Phys.

Hiroshima Math. Journ.

Schrödinger equation for the system of one-dimensional particles with point interaction

Some remarks about the theory of spherical functions on a symmetric space

Zur Theorie der Metalle, I. Eigenwerte und Eigenfunktionen der linearen Atomkette

Zs. Phys.

Un probleme à N corps soluble

Compt. Rend. Acad. Sci. Paris

Ground state of a one-dimensional N-body system

J. Math. Phys.

Comparison between the exact and Hartree solutions of a one-dimensional many-body problem

Phys. Rev.

Exact solution of a one-dimensional three-body scattering problem with two-body and/or three-body inverse-square potentials

J. Math. Phys.

Exact solution of the classical and quantal one-dimensional many-body problems with the two-body potential Va(x) = g2a2sinh−2 ax

Lett. Nuovo Cim.

Regular polytopes

Quantum completely integrable models in field theory

Soviet Scientific Reviews, Math. Phys.

Comparison of exact quantum and quasiclassical answers for nonlinear Schrödinger equation

TMF

Exact results in quantum many-body systems of interacting particles in many dimensions with SU(1,1) as the dynamical group

J. Math. Phys.

Binding energy of a Bose gas in one dimension

Phys. Rev.

Spherical functions on symmetric Riemann spaces

Dokl. Akad. Nauk SSSR

Unitary representations of the classical groups

Plancherel measure of Riemannian symmetric spaces of nonpositive curvature

Soviet Math. Dokl.

Unitary representations of automorphism groups of symmetric spaces with zero curvature

Funk. Anal. Appl

Completely integrable systems and groups generated by reflections

Short proof of a conjecture of Dyson

J. Math. Phys.

Spherical functions on a semi-simple Lie group I, II

Amer. J. Math.

Spherical functions on a semi-simple Lie group I, II

Amer. J. Math.

Differential Geometry, Lie Groups and Symmetric Spaces

Simple irreducible graded Lie algebras of finite growth

Izv. Acad. Nauk SSSR ser. Math.

Geometry of geodesics and eigenfunctions of the Laplace-Beltrami operator

Quantization and representation theory of Lie groups

Factorization of classical and quantum S-matrices and conservation laws

Theor. Math. Phys.

Spherically symmetric equations in gauge theories for an arbitrary semisimple compact Lie group

Phys. Lett.

Exact analysis of an interacting Bose gas, I. The general solution and the ground state

Phys. Rev.

Affine root systems and Dedekind's η-function

Invent. Math.

Some conjectures for root systems

SIAM Journ. Math. Anal.

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J. Math. Phys.

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J. Algebra

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Uspekhi Math. Nauk

Quantum completely integrable systems connected with semi-simple Lie algebras

Lett. Math. Phys.

Quantum systems connected with root systems and the radial parts of Laplace operators

Funct. Anal. Appl.

Exact results for one-dimensional many-body systems

Fisica elementarnikh chastiz i atomnogo yadra

Integrability and a solution for the one-dimensional N-particle system with inversely quadratic pair potentials

J. Phys. Soc. Japan

Exact solution of the classical and quantal one-dimensional many-body problems with the two-body potential V_a(x) = g²a²sinh⁻² ax