Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-th/9906157

Quantum Davey-Stewartson 1 System with Two Components in Two Dimensions

Mu-Lin Yan

Published Jun 21, 1999• 0 citations
arXivHTMLPDF

Abstract

The quantum 2-component DS1 system in two dimensions has been reduced to two 2-component many-body problems both in one dimension by an analogue of the method of separation of variables. The solutions to 2 dimensional problem under consideration can be constructed from the resulting problem in one dimensions. For latters with the $\\delta$-function interactions and being solved by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system.

Related papers

Quantum integrable system with two color components in two dimensions

Mu-Lin Yan

The Davey-Stewartson 1 (DS1) system [Proc. R. Soc. London, Ser. A 338, 101 (1974)] is an integrable model in two dimensions. A quantum DS1 system with two color components in two dimensions has been reformulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with two color components. The solutions to the two-dimensional problem under consideration can be constructed from the resulting problem in one dimensions. For the latter problem with delta-function interactions and solution by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system.

arXiv:cond-mat/0111427•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 1, 2000•0 citations

Bethe Ansatz and Quantum Davey-Stewartson 1 System with Multicomponent in Two Dimensions

Yi Cheng, Mu-Lin Yan, Bao‐Heng Zhao

The quantum 2-component DS1 system was reduced to two 1D many-body problems with $δ-$function interactions, which were solved by Bethe ansatz. Using the ansatz in ref.[1] and introducing symmetric and antisymmetric Young operators of the permutation group, we obtain the exact solutions for the system.

arXiv:cond-mat/9402086•Feb 21, 1994•0 citations

QUANTUM INTEGRABLE SYSTEM WITH MULTI-COMPONENTS IN TWO DIMENSIONS

Mu-Lin Yan, Bao‐Heng Zhao

A quantum N-body problem with 2-component in (2 + 1) dimensions deduced from an integrable model in (2 + 1) dimensions is investigated. The Davey–Stewartson 1 (DS1) system9 is an integrable model in two dimensions. A quantum DS1 system with 2 color components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 color components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in one dimension. For the latter with δ-function interactions and being solved by the Bethe–Yang ansatz, we introduce symmetrical and anti-symmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system. The application of the solutions is discussed.

arXiv:cond-mat/0209036•Modern Physics Letters B•Oct 20, 2002•0 citations

Analysis of the Many-Body Problem in One Dimension with Repulsive Delta-Function Interaction

M. Albertsson

The repulsive delta-function interaction model in one dimension is reviewed for spinless particles and for spin-1/2 fermions. The problem of solving the differential equation related to the Schrödinger equation is reduced by the Bethe ansatz to a system of algebraic equations. The delta-function interaction is shown to have no effect on spinless fermions which therefore behave like free fermions, in agreement with Pauli's exclusion principle. The ground-state problem of spinless bosons is reduced to an inhomogeneous Fredholm equation of the second kind. In the limit of impenetrable interactions, the spinless bosons are shown to have the energy spectrum of free fermions. The model for spin-1/2 fermions is reduced by the Bethe ansatz to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group of N elements. For some R's this eigenvalue problem itself is solved by a generalized Bethe ansatz. The ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation.

•Jan 1, 2014•1 citation

Quantum many-body systems exactly solved by special functions

Martin A. Hallnäs

This thesis concerns two types of quantum many-body systems in one dimension exactly solved by special functions: firstly, systems with interactions localised at points and solved by the (coordinate) Bethe ansatz; secondly, systems of Calogero-Sutherland type, as well as certain recently introduced deformations thereof, with eigenfunctions given by natural many-variable generalisations of classical (orthogonal) polynomials. The thesis is divided into two parts. The first provides background and a few complementary results, while the second presents the main results of this thesis in five appended scientific papers. In the first paper we consider two complementary quantum many-body systems with local interactions related to the root systems CN, one with delta-interactions, and the other with certain momentum dependent interactions commonly known as delta-prime interactions. We prove, by construction, that the former is exactly solvable by the Bethe ansatz in the general case of distinguishable particles, and that the latter is similarly solvable only in the case of bosons or fermions. We also establish a simple strong/weak coupling duality between the two models and elaborate on their physical interpretations. In the second paper we consider a well-known four-parameter family of local interactions in one dimension. In particular, we determine all such interactions leading to a quantum many-body system of distinguishable particles exactly solvable by the Bethe ansatz. We find that there are two families of such systems: the first is described by a one-parameter deformation of the delta-interaction model, while the second features a particular one-parameter combination of the delta and the delta-prime interactions. In papers 3-5 we construct and study particular series representations for the eigenfunctions of a family of Calogero-Sutherland models naturally associated with the classical (orthogonal) polynomials. In our construction, the eigenfunctions are given by linear combinations of certain symmetric polynomials generalising the so-called Schur polynomials, with explicit and rather simple coefficients. In paper 5 we also generalise certain of these results to the so-called deformed Calogero-Sutherland operators.

•Jan 1, 2007•1 citation

Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction

Chongming Yang

The repulsive $\ensuremath{\delta}$ interaction problem in one dimension for $N$ particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations $R$ of the permutation group ${S}_{N}$. For some $R'\mathrm{s}$ this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.

Physical Review Letters•Dec 4, 1967•2,402 citations
The quantum 2-component DS1 system in two dimensions has been reduced to two 2-component many-body problems both in one dimension by an analogue of the method of separation of variables. The solutions to 2 dimensional problem under consideration can be constructed from the resulting problem in one dimensions. For latters with the delta\\deltadelta-function interactions and being solved by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system.
The Davey-Stewartson 1 (DS1) system [Proc. R. Soc. London, Ser. A 338, 101 (1974)] is an integrable model in two dimensions. A quantum DS1 system with two color components in two dimensions has been reformulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with two color components. The solutions to the two-dimensional problem under consideration can be constructed from the resulting problem in one dimensions. For the latter problem with delta-function interactions and solution by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system.
The quantum 2-component DS1 system was reduced to two 1D many-body problems with δ−δ-δ−function interactions, which were solved by Bethe ansatz. Using the ansatz in ref.[1] and introducing symmetric and antisymmetric Young operators of the permutation group, we obtain the exact solutions for the system.
A quantum N-body problem with 2-component in (2 + 1) dimensions deduced from an integrable model in (2 + 1) dimensions is investigated. The Davey–Stewartson 1 (DS1) system9 is an integrable model in two dimensions. A quantum DS1 system with 2 color components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 color components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in one dimension. For the latter with δ-function interactions and being solved by the Bethe–Yang ansatz, we introduce symmetrical and anti-symmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system. The application of the solutions is discussed.
The repulsive delta-function interaction model in one dimension is reviewed for spinless particles and for spin-1/2 fermions. The problem of solving the differential equation related to the Schrödinger equation is reduced by the Bethe ansatz to a system of algebraic equations. The delta-function interaction is shown to have no effect on spinless fermions which therefore behave like free fermions, in agreement with Pauli's exclusion principle. The ground-state problem of spinless bosons is reduced to an inhomogeneous Fredholm equation of the second kind. In the limit of impenetrable interactions, the spinless bosons are shown to have the energy spectrum of free fermions. The model for spin-1/2 fermions is reduced by the Bethe ansatz to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group of N elements. For some R's this eigenvalue problem itself is solved by a generalized Bethe ansatz. The ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation.
This thesis concerns two types of quantum many-body systems in one dimension exactly solved by special functions: firstly, systems with interactions localised at points and solved by the (coordinate) Bethe ansatz; secondly, systems of Calogero-Sutherland type, as well as certain recently introduced deformations thereof, with eigenfunctions given by natural many-variable generalisations of classical (orthogonal) polynomials. The thesis is divided into two parts. The first provides background and a few complementary results, while the second presents the main results of this thesis in five appended scientific papers. In the first paper we consider two complementary quantum many-body systems with local interactions related to the root systems CN, one with delta-interactions, and the other with certain momentum dependent interactions commonly known as delta-prime interactions. We prove, by construction, that the former is exactly solvable by the Bethe ansatz in the general case of distinguishable particles, and that the latter is similarly solvable only in the case of bosons or fermions. We also establish a simple strong/weak coupling duality between the two models and elaborate on their physical interpretations. In the second paper we consider a well-known four-parameter family of local interactions in one dimension. In particular, we determine all such interactions leading to a quantum many-body system of distinguishable particles exactly solvable by the Bethe ansatz. We find that there are two families of such systems: the first is described by a one-parameter deformation of the delta-interaction model, while the second features a particular one-parameter combination of the delta and the delta-prime interactions. In papers 3-5 we construct and study particular series representations for the eigenfunctions of a family of Calogero-Sutherland models naturally associated with the classical (orthogonal) polynomials. In our construction, the eigenfunctions are given by linear combinations of certain symmetric polynomials generalising the so-called Schur polynomials, with explicit and rather simple coefficients. In paper 5 we also generalise certain of these results to the so-called deformed Calogero-Sutherland operators.
The repulsive $\ensuremath{\delta}$ interaction problem in one dimension for NNN particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations RRR of the permutation group SN{S}_{N}SN​. For some R′sR'\mathrm{s}R′s this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.