Quantum integrable systems related to lie algebras
Identifieur interne : 002840 ( Main/Merge ); précédent : 002839; suivant : 002841Quantum integrable systems related to lie algebras
Auteurs : M. A. Olshanetsky [Russie] ; A. M. Perelomov [Russie]Source :
- Physics Reports [ 0370-1573 ] ; 1983.
English descriptors
- KwdEn :
- Abstract case, Additional term, Adjoint representation, Affine, Affine root system, Affine root systems, Affine weyl group, Airy function, Alcove, Algebra, Ansatz, Asymptotic, Asymptotic behaviour, Automorphism, Bethe ansatz, Billiards problem, Boundary conditions, Calogero, Cartan, Cartan decomposition, Cartan subalgebra, Classical integrals, Classical type, Coefficient, Commutator, Complete integrability, Complete list, Complex group, Complex systems, Configuration space, Conjecture, Cosh, Coxeter, Coxeter group, Coxeter systems, Curvature, Determinant, Diagonal matrix, Dihedral group, Dynkin, Dynkin graph, Dynkin graphs, Dyson, Eigenfunction, Eigenfunctions, Eigenvalue, Eigenvalue problem, Energy levels, Energy spectrum, Euclidean space, Exact results, Explicit expression, Explicit expressions, Explicit form, Factorization, Free motion, General form, General solution, Gradient root system, Ground state, Group case, Hamiltonian, Horospheric, Horospheric projection, Hyperplane, Integrability, Integrable, Integrable systems, Integral, Integral representation, Involutive, Involutive automorphism, Irreducible, Jacobi, Jacobi polynomials, Laplace, Laplace operator, Laplace operators, Lattice, Lett, Little subgroup, Math, Matrix, Maximal root, Minimal root, Nauk sssr, Negative curvature, Nilpotent subgroup, Noncrystallographic, Nonpositive curvature, Nonreduced, Normal form, Normalization, Obtains, Olshanetsky, Other hand, Pairwise, Particular case, Perelomov, Periodic toda lattice, Permutation, Permutation group, Phys, Physics reports, Plane waves, Polynomial solutions, Positive curvature, Positive roots, Quantum, Quantum field theory, Quantum iniegrable systems, Quantum integrable systems, Quantum integrals, Quantum systems, Radial part, Radial parts, Repulsive case, Root subspaces, Root system, Root systems, Schrodinger, Schrodinger equation, Selfadjoint, Semisimple, Simple roots, Sin2, Sinh, Sinh2, Special properties, Spherical functions, Statistical theory, Subalgebra, Subgroup, Subsystem, Such systems, Symmetric, Symmetric pair, Symmetric space, Symmetric spaces, Toda, Toda lattice, Toda lattices, Unperiodic toda lattice, Wave function, Wave functions, Weierstrass, Weierstrass function, Weight lattice, Weyl, Weyl alcove, Weyl alcoves, Weyl chamber, Weyl chambers, Weyl group, Zonal.
- Teeft :
- Abstract case, Additional term, Adjoint representation, Affine, Affine root system, Affine root systems, Affine weyl group, Airy function, Alcove, Algebra, Ansatz, Asymptotic, Asymptotic behaviour, Automorphism, Bethe ansatz, Billiards problem, Boundary conditions, Calogero, Cartan, Cartan decomposition, Cartan subalgebra, Classical integrals, Classical type, Coefficient, Commutator, Complete integrability, Complete list, Complex group, Complex systems, Configuration space, Conjecture, Cosh, Coxeter, Coxeter group, Coxeter systems, Curvature, Determinant, Diagonal matrix, Dihedral group, Dynkin, Dynkin graph, Dynkin graphs, Dyson, Eigenfunction, Eigenfunctions, Eigenvalue, Eigenvalue problem, Energy levels, Energy spectrum, Euclidean space, Exact results, Explicit expression, Explicit expressions, Explicit form, Factorization, Free motion, General form, General solution, Gradient root system, Ground state, Group case, Hamiltonian, Horospheric, Horospheric projection, Hyperplane, Integrability, Integrable, Integrable systems, Integral, Integral representation, Involutive, Involutive automorphism, Irreducible, Jacobi, Jacobi polynomials, Laplace, Laplace operator, Laplace operators, Lattice, Lett, Little subgroup, Math, Matrix, Maximal root, Minimal root, Nauk sssr, Negative curvature, Nilpotent subgroup, Noncrystallographic, Nonpositive curvature, Nonreduced, Normal form, Normalization, Obtains, Olshanetsky, Other hand, Pairwise, Particular case, Perelomov, Periodic toda lattice, Permutation, Permutation group, Phys, Physics reports, Plane waves, Polynomial solutions, Positive curvature, Positive roots, Quantum, Quantum field theory, Quantum iniegrable systems, Quantum integrable systems, Quantum integrals, Quantum systems, Radial part, Radial parts, Repulsive case, Root subspaces, Root system, Root systems, Schrodinger, Schrodinger equation, Selfadjoint, Semisimple, Simple roots, Sin2, Sinh, Sinh2, Special properties, Spherical functions, Statistical theory, Subalgebra, Subgroup, Subsystem, Such systems, Symmetric, Symmetric pair, Symmetric space, Symmetric spaces, Toda, Toda lattice, Toda lattices, Unperiodic toda lattice, Wave function, Wave functions, Weierstrass, Weierstrass function, Weight lattice, Weyl, Weyl alcove, Weyl alcoves, Weyl chamber, Weyl chambers, Weyl group, Zonal.
Abstract
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: vI(q) = q−2, vII(q) = sinh−2 q, vIII(q) = sin−2 q, vIV(q) = P(q), vV(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(qjqj+ 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.
Url:
DOI: 10.1016/0370-1573(83)90018-2
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<profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Abstract case</term>
<term>Additional term</term>
<term>Adjoint representation</term>
<term>Affine</term>
<term>Affine root system</term>
<term>Affine root systems</term>
<term>Affine weyl group</term>
<term>Airy function</term>
<term>Alcove</term>
<term>Algebra</term>
<term>Ansatz</term>
<term>Asymptotic</term>
<term>Asymptotic behaviour</term>
<term>Automorphism</term>
<term>Bethe ansatz</term>
<term>Billiards problem</term>
<term>Boundary conditions</term>
<term>Calogero</term>
<term>Cartan</term>
<term>Cartan decomposition</term>
<term>Cartan subalgebra</term>
<term>Classical integrals</term>
<term>Classical type</term>
<term>Coefficient</term>
<term>Commutator</term>
<term>Complete integrability</term>
<term>Complete list</term>
<term>Complex group</term>
<term>Complex systems</term>
<term>Configuration space</term>
<term>Conjecture</term>
<term>Cosh</term>
<term>Coxeter</term>
<term>Coxeter group</term>
<term>Coxeter systems</term>
<term>Curvature</term>
<term>Determinant</term>
<term>Diagonal matrix</term>
<term>Dihedral group</term>
<term>Dynkin</term>
<term>Dynkin graph</term>
<term>Dynkin graphs</term>
<term>Dyson</term>
<term>Eigenfunction</term>
<term>Eigenfunctions</term>
<term>Eigenvalue</term>
<term>Eigenvalue problem</term>
<term>Energy levels</term>
<term>Energy spectrum</term>
<term>Euclidean space</term>
<term>Exact results</term>
<term>Explicit expression</term>
<term>Explicit expressions</term>
<term>Explicit form</term>
<term>Factorization</term>
<term>Free motion</term>
<term>General form</term>
<term>General solution</term>
<term>Gradient root system</term>
<term>Ground state</term>
<term>Group case</term>
<term>Hamiltonian</term>
<term>Horospheric</term>
<term>Horospheric projection</term>
<term>Hyperplane</term>
<term>Integrability</term>
<term>Integrable</term>
<term>Integrable systems</term>
<term>Integral</term>
<term>Integral representation</term>
<term>Involutive</term>
<term>Involutive automorphism</term>
<term>Irreducible</term>
<term>Jacobi</term>
<term>Jacobi polynomials</term>
<term>Laplace</term>
<term>Laplace operator</term>
<term>Laplace operators</term>
<term>Lattice</term>
<term>Lett</term>
<term>Little subgroup</term>
<term>Math</term>
<term>Matrix</term>
<term>Maximal root</term>
<term>Minimal root</term>
<term>Nauk sssr</term>
<term>Negative curvature</term>
<term>Nilpotent subgroup</term>
<term>Noncrystallographic</term>
<term>Nonpositive curvature</term>
<term>Nonreduced</term>
<term>Normal form</term>
<term>Normalization</term>
<term>Obtains</term>
<term>Olshanetsky</term>
<term>Other hand</term>
<term>Pairwise</term>
<term>Particular case</term>
<term>Perelomov</term>
<term>Periodic toda lattice</term>
<term>Permutation</term>
<term>Permutation group</term>
<term>Phys</term>
<term>Physics reports</term>
<term>Plane waves</term>
<term>Polynomial solutions</term>
<term>Positive curvature</term>
<term>Positive roots</term>
<term>Quantum</term>
<term>Quantum field theory</term>
<term>Quantum iniegrable systems</term>
<term>Quantum integrable systems</term>
<term>Quantum integrals</term>
<term>Quantum systems</term>
<term>Radial part</term>
<term>Radial parts</term>
<term>Repulsive case</term>
<term>Root subspaces</term>
<term>Root system</term>
<term>Root systems</term>
<term>Schrodinger</term>
<term>Schrodinger equation</term>
<term>Selfadjoint</term>
<term>Semisimple</term>
<term>Simple roots</term>
<term>Sin2</term>
<term>Sinh</term>
<term>Sinh2</term>
<term>Special properties</term>
<term>Spherical functions</term>
<term>Statistical theory</term>
<term>Subalgebra</term>
<term>Subgroup</term>
<term>Subsystem</term>
<term>Such systems</term>
<term>Symmetric</term>
<term>Symmetric pair</term>
<term>Symmetric space</term>
<term>Symmetric spaces</term>
<term>Toda</term>
<term>Toda lattice</term>
<term>Toda lattices</term>
<term>Unperiodic toda lattice</term>
<term>Wave function</term>
<term>Wave functions</term>
<term>Weierstrass</term>
<term>Weierstrass function</term>
<term>Weight lattice</term>
<term>Weyl</term>
<term>Weyl alcove</term>
<term>Weyl alcoves</term>
<term>Weyl chamber</term>
<term>Weyl chambers</term>
<term>Weyl group</term>
<term>Zonal</term>
</keywords>
<keywords scheme="Teeft" xml:lang="en"><term>Abstract case</term>
<term>Additional term</term>
<term>Adjoint representation</term>
<term>Affine</term>
<term>Affine root system</term>
<term>Affine root systems</term>
<term>Affine weyl group</term>
<term>Airy function</term>
<term>Alcove</term>
<term>Algebra</term>
<term>Ansatz</term>
<term>Asymptotic</term>
<term>Asymptotic behaviour</term>
<term>Automorphism</term>
<term>Bethe ansatz</term>
<term>Billiards problem</term>
<term>Boundary conditions</term>
<term>Calogero</term>
<term>Cartan</term>
<term>Cartan decomposition</term>
<term>Cartan subalgebra</term>
<term>Classical integrals</term>
<term>Classical type</term>
<term>Coefficient</term>
<term>Commutator</term>
<term>Complete integrability</term>
<term>Complete list</term>
<term>Complex group</term>
<term>Complex systems</term>
<term>Configuration space</term>
<term>Conjecture</term>
<term>Cosh</term>
<term>Coxeter</term>
<term>Coxeter group</term>
<term>Coxeter systems</term>
<term>Curvature</term>
<term>Determinant</term>
<term>Diagonal matrix</term>
<term>Dihedral group</term>
<term>Dynkin</term>
<term>Dynkin graph</term>
<term>Dynkin graphs</term>
<term>Dyson</term>
<term>Eigenfunction</term>
<term>Eigenfunctions</term>
<term>Eigenvalue</term>
<term>Eigenvalue problem</term>
<term>Energy levels</term>
<term>Energy spectrum</term>
<term>Euclidean space</term>
<term>Exact results</term>
<term>Explicit expression</term>
<term>Explicit expressions</term>
<term>Explicit form</term>
<term>Factorization</term>
<term>Free motion</term>
<term>General form</term>
<term>General solution</term>
<term>Gradient root system</term>
<term>Ground state</term>
<term>Group case</term>
<term>Hamiltonian</term>
<term>Horospheric</term>
<term>Horospheric projection</term>
<term>Hyperplane</term>
<term>Integrability</term>
<term>Integrable</term>
<term>Integrable systems</term>
<term>Integral</term>
<term>Integral representation</term>
<term>Involutive</term>
<term>Involutive automorphism</term>
<term>Irreducible</term>
<term>Jacobi</term>
<term>Jacobi polynomials</term>
<term>Laplace</term>
<term>Laplace operator</term>
<term>Laplace operators</term>
<term>Lattice</term>
<term>Lett</term>
<term>Little subgroup</term>
<term>Math</term>
<term>Matrix</term>
<term>Maximal root</term>
<term>Minimal root</term>
<term>Nauk sssr</term>
<term>Negative curvature</term>
<term>Nilpotent subgroup</term>
<term>Noncrystallographic</term>
<term>Nonpositive curvature</term>
<term>Nonreduced</term>
<term>Normal form</term>
<term>Normalization</term>
<term>Obtains</term>
<term>Olshanetsky</term>
<term>Other hand</term>
<term>Pairwise</term>
<term>Particular case</term>
<term>Perelomov</term>
<term>Periodic toda lattice</term>
<term>Permutation</term>
<term>Permutation group</term>
<term>Phys</term>
<term>Physics reports</term>
<term>Plane waves</term>
<term>Polynomial solutions</term>
<term>Positive curvature</term>
<term>Positive roots</term>
<term>Quantum</term>
<term>Quantum field theory</term>
<term>Quantum iniegrable systems</term>
<term>Quantum integrable systems</term>
<term>Quantum integrals</term>
<term>Quantum systems</term>
<term>Radial part</term>
<term>Radial parts</term>
<term>Repulsive case</term>
<term>Root subspaces</term>
<term>Root system</term>
<term>Root systems</term>
<term>Schrodinger</term>
<term>Schrodinger equation</term>
<term>Selfadjoint</term>
<term>Semisimple</term>
<term>Simple roots</term>
<term>Sin2</term>
<term>Sinh</term>
<term>Sinh2</term>
<term>Special properties</term>
<term>Spherical functions</term>
<term>Statistical theory</term>
<term>Subalgebra</term>
<term>Subgroup</term>
<term>Subsystem</term>
<term>Such systems</term>
<term>Symmetric</term>
<term>Symmetric pair</term>
<term>Symmetric space</term>
<term>Symmetric spaces</term>
<term>Toda</term>
<term>Toda lattice</term>
<term>Toda lattices</term>
<term>Unperiodic toda lattice</term>
<term>Wave function</term>
<term>Wave functions</term>
<term>Weierstrass</term>
<term>Weierstrass function</term>
<term>Weight lattice</term>
<term>Weyl</term>
<term>Weyl alcove</term>
<term>Weyl alcoves</term>
<term>Weyl chamber</term>
<term>Weyl chambers</term>
<term>Weyl group</term>
<term>Zonal</term>
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<front><div type="abstract" xml:lang="en">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: vI(q) = q−2, vII(q) = sinh−2 q, vIII(q) = sin−2 q, vIV(q) = P(q), vV(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(qjqj+ 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.</div>
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