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arXiv:1109.6604

Higher conservation laws for the quantum non-linear Schroedinger equation

Brian Davies, V. E. Korepin

Published Sep 29, 2011• 21 citations
arXivHTMLPDF

Abstract

Quantum non-linear SCHROEDINGER equation is equivalent to Lieb-Liniger model. It has non-trivial conservation laws. Recently these conservation laws were used for evaluation of the three-body recombination rate for interacting gas of quantum bosons. These conservations laws were known already in 1989. Submitted text is retyping of the preprint of Centre for Mathematical Analysis of Australian National University CMA-R33-89. It was discussed in Leningrad Branch of the V.A. Steklov Mathematical Institute at that time. A copy of the original preprint can be found in the section Quantum Inverse Scattering Method of the web-page http://insti.physics.sunysb.edu/~korepin/

DOI
10.48550/arxiv.1109.6604

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Quantum non-linear SCHROEDINGER equation is equivalent to Lieb-Liniger model. It has non-trivial conservation laws. Recently these conservation laws were used for evaluation of the three-body recombination rate for interacting gas of quantum bosons. These conservations laws were known already in 1989. Submitted text is retyping of the preprint of Centre for Mathematical Analysis of Australian National University CMA-R33-89. It was discussed in Leningrad Branch of the V.A. Steklov Mathematical Institute at that time. A copy of the original preprint can be found in the section Quantum Inverse Scattering Method of the web-page http://insti.physics.sunysb.edu/~korepin/
The generalisation of the quantum inverse scattering method to the study of direct and inverse problems for the multicomponent non-linear Schrodinger model of bosons or fermions with repulsive coupling is made. Two sets of Yang-Baxter equations are solved to obtain commutation relations between the scattering state operators. The eigenfunctions have been constructed for the infinite number of conserved quantities and the eigenvalues of the first three conserved quantities-number of particles, momentum and energy-are obtained. The global Izergin-Korepin relations and the relations between the quantum Jost functions are derived, and from them, the quantum Gel'fand-Levitan equations are established. Finally, the series expansion for the field operators in terms of the scattering state operators is written out explicitly.
We consider the quantum nonlinear Schrodinger equation in one space and one time dimension. It is equivalent to Bose gas with delta interaction. We are interested in the non-free-fermionic case. We calculate temperature-dependent correlation functions. The determinant representation for correlation functions simplifies in the small mass limit of the Bose particle. In this limit we describe the correlation function in terms of the Maxwell-Bloch differential equation. We solve this equation at large distance and evaluate asymptotics of correlation functions. 1 kojima@kurims.kyoto-u.ac.jp 2 korepin@yukawa.kyoto-u.ac.jp 2 korepin@insti.physics.sunysb.edu ? Research Fellow of the Japan Society for the Promotion of Science. 1 Introduction In this paper we consider correlation functions of exactly solvable models. Our approach is based on the determinant representation of quantum correlation functions [1]. We consider the thermodynamics of Bose gas with delta-interaction at finite temper...
This chapter explores multi-body scattering in quantum integrable models, emphasizing the emergence of Factorized Scattering Theory. The presence of an extensive set of conservation laws ensures the non-diffractive nature of scattering, leading to a distinctive form of the asymptotic wave function – known as the Bethe wave function – as a linear superposition of plane waves. Elementary scattering events, such as reflection and transmission, are described by Yang’s scattering operators and the two-body S -matrix. The multi-body S -matrix is then expressed as an ordered product of two-body S -matrices, with order-independence guaranteed by the Yang–Baxter equation. To illustrate these ideas, the Lieb–Liniger model for distinguishable particles is introduced. The chapter also examines general symmetry constraints on the Bethe wave function imposed by quantum statistics.