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:quant-ph/0205145 · Birkhäuser Basel eBooks

On Integrability of Many-body Problems with Point Interactions

Sergio A Albeverio, S. M. Fei, Pavel Kurasov

Published Jan 1, 2002• 6 citations
arXivHTMLPDF

Abstract

A study of the integrability of one-dimensional quantum mechanical many-body systems with general point interactions and boundary conditions describing the interactions which can be independent or dependent on the spin states of the particles is presented. The corresponding Bethe ansatz solutions, bound states and scattering matrices are explicitly given. Hamilton operators corresponding to special spin dependent boundary conditions are discussed.

DOI
10.1007/978-3-0348-8219-4_7

Related papers

A REMARK ON ONE-DIMENSIONAL MANY-BODY PROBLEMS WITH POINT INTERACTIONS

Sergio A Albeverio, L. Dąbrowski, Shao-Ming Fei

The integrability of one-dimensional quantum mechanical many-body problems with general contact interactions is extensively studied. It is shown that besides the pure (repulsive or attractive) δ-function interaction there is another singular point interactions which gives rise to a new one-parameter family of integrable quantum mechanical many-body systems. The bound states and scattering matrices are calculated for both bosonic and fermionic statistics.

arXiv:quant-ph/0001089•International Journal of Modern Physics B•Mar 20, 2000•14 citations

Boundary Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-Dependent Bulk and Boundary Coupling Strengths

Parameshwar R. Pasnoori

The generalized Bethe ansatz framework formulated in [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)] provides a unified framework to find exact solutions to quantum many-body systems with time-dependent coupling strengths with periodic boundary conditions. In this work we extend this framework to the case of open boundary conditions where in addition to the time-dependent interactions in the bulk, the boundary conditions are explicitly time-dependent. We show that for integrable time-dependent bulk coupling strengths, the generalized Bethe ansatz framework provides the time-dependent boundary conditions compatible with integrability and reduces the time-dependent Schrodinger equation to a set of matrix difference equations called the boundary quantum Knizhnik-Zamolodchikov (BqKZ) equations. The solution to the BqKZ equations provides the explicit form of the exact wavefunction. We further show that the RG invariants of the corresponding static model identify with the dynamical invariants in the time-dependent model.

math-phCross-listedarXiv:2608.31080•Sep 1, 2026•0 citations

On the Quantum Integrability for A Model of Coupled XY Chains

Huan-Qiang Zhou, Xiangyu Ge

The paper discusses integrable open-boundary conditions for the model of three coupled one-dimensional XY spin chains in the framework of the quantum inverse scattering method. The diagonal boundary K-matrices are found and a class of integrable boundary terms are determined. The boundary model Hamiltonian is solved by using the coordinate space Bethe ansatz technique and Bathe ansatz equations are derived.

•Jan 1, 1999•0 citations

Aspects of Integrable Models

Pranav Diwakar

The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz. To this end, we discuss the concept of integrability as well as the Lax operator and R-matrix, which help generate as many commuting operators in involution as there are degrees of freedom. We establish that the spin chain Hamiltonian belongs to this set and provide a definition of a state vector whose parameters, the Bethe roots, are constrained by a set of equations called the Bethe Ansatz Equations. We show that there is a one-to-one correspondence between the Bethe roots and the eigenfunctions of the system. Next, we proceed to study the nature of the low-lying excitations of both the ferromagnetic and antiferromagnetic model in the thermodynamic limit N → ∞ and show that the Bethe roots can be grouped into complexes or strings, which behave like bound states. We see that integrability is directly related to diffractionless scattering, which is obeyed by systems whose scattering matrices satisfy the Yang-Baxter Equation. In order to provide a more physical interpretation, we calculate the scattering matrix of the two-body problem for a system that satisfies the Yang-Baxter Equation and obtain exchange relations that are identical to those obtained using the Algebraic Bethe Ansatz for the XXX-1/2 spin chain model. Finally, we calculate the scattering matrix for a two-body problem interacting with a delta potential and show that this is the same as what we derived using the Coordinate Bethe Ansatz.

•Oct 18, 2018•0 citations

Spin-dependent point potentials in one and three dimensions

Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari

We consider a system realized with one spinless quantum particle and an array of N spins 1/2 in dimensions 1 and 3. We characterize all the Hamiltonians obtained as point perturbations of assigned free dynamics in terms of some generalized boundary conditions. For every boundary condition, we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin-dependent point potentials that could be of interest as multi-component solvable models.

arXiv:math-ph/0609055•Journal of Physics A Mathematical and Theoretical•Dec 12, 2006•26 citations

Classical facets of quantum integrability

A. Zabrodin

This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational $GL(n)$-invariant $R$-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.

arXiv:2501.18557•Jan 30, 2025•0 citations
A study of the integrability of one-dimensional quantum mechanical many-body systems with general point interactions and boundary conditions describing the interactions which can be independent or dependent on the spin states of the particles is presented. The corresponding Bethe ansatz solutions, bound states and scattering matrices are explicitly given. Hamilton operators corresponding to special spin dependent boundary conditions are discussed.
The generalized Bethe ansatz framework formulated in [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)] provides a unified framework to find exact solutions to quantum many-body systems with time-dependent coupling strengths with periodic boundary conditions. In this work we extend this framework to the case of open boundary conditions where in addition to the time-dependent interactions in the bulk, the boundary conditions are explicitly time-dependent. We show that for integrable time-dependent bulk coupling strengths, the generalized Bethe ansatz framework provides the time-dependent boundary conditions compatible with integrability and reduces the time-dependent Schrodinger equation to a set of matrix difference equations called the boundary quantum Knizhnik-Zamolodchikov (BqKZ) equations. The solution to the BqKZ equations provides the explicit form of the exact wavefunction. We further show that the RG invariants of the corresponding static model identify with the dynamical invariants in the time-dependent model.
The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz. To this end, we discuss the concept of integrability as well as the Lax operator and R-matrix, which help generate as many commuting operators in involution as there are degrees of freedom. We establish that the spin chain Hamiltonian belongs to this set and provide a definition of a state vector whose parameters, the Bethe roots, are constrained by a set of equations called the Bethe Ansatz Equations. We show that there is a one-to-one correspondence between the Bethe roots and the eigenfunctions of the system. Next, we proceed to study the nature of the low-lying excitations of both the ferromagnetic and antiferromagnetic model in the thermodynamic limit N → ∞ and show that the Bethe roots can be grouped into complexes or strings, which behave like bound states. We see that integrability is directly related to diffractionless scattering, which is obeyed by systems whose scattering matrices satisfy the Yang-Baxter Equation. In order to provide a more physical interpretation, we calculate the scattering matrix of the two-body problem for a system that satisfies the Yang-Baxter Equation and obtain exchange relations that are identical to those obtained using the Algebraic Bethe Ansatz for the XXX-1/2 spin chain model. Finally, we calculate the scattering matrix for a two-body problem interacting with a delta potential and show that this is the same as what we derived using the Coordinate Bethe Ansatz.
We consider a system realized with one spinless quantum particle and an array of N spins 1/2 in dimensions 1 and 3. We characterize all the Hamiltonians obtained as point perturbations of assigned free dynamics in terms of some generalized boundary conditions. For every boundary condition, we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin-dependent point potentials that could be of interest as multi-component solvable models.
This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational GL(n)GL(n)GL(n)-invariant RRR-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.