# The Library

### Norm formulae for the Bethe Ansatz on root systems of small rank

Tools

Bustamante, Miguel D., Van Diejen, Jan Felipe, 1965- and de la Maza, A. C..
(2008)
*Norm formulae for the Bethe Ansatz on root systems of small rank.*
Journal of Physics A: Mathematical and Theoretical, Volume 41
(Number 2).
Article No. 025202.
ISSN 1751-8113

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1088/1751-8113/41/2/025202

## Abstract

The norms of the Bethe Ansatz eigenfunctions for the Lieb-Liniger quantum system of n Bosonic particles on a ring with pairwise repulsive delta potential interactions are given by a beautiful determinantal formula, first conjectured by Gaudin in the early seventies and then proven by Korepin about a decade later. Recently, E Emsiz formulated a similar conjecture generalizing the Gaudin-Korepin norm formula in terms of the root systems of complex simple Lie algebras. Here we confirm the validity of the conjecture in question for small root systems up to rank 3 ( thus including the important test case of the exceptional root system G(2)).

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics Q Science > QC Physics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Bethe-ansatz technique, Root systems (Algebra), Eigenfunctions |

Journal or Publication Title: | Journal of Physics A: Mathematical and Theoretical |

Publisher: | IOP Publishing Ltd |

ISSN: | 1751-8113 |

Date: | 18 January 2008 |

Volume: | Volume 41 |

Number: | Number 2 |

Number of Pages: | 13 |

Page Range: | Article No. 025202 |

Identification Number: | 10.1088/1751-8113/41/2/025202 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | Fondo Nacional de Desarrollo Científico y Tecnológico (Chile) (FONDECYT), World Bank, Universidad de Talca |

Grant number: | 1051012, 1040896 (FONDECYT) |

References: | [B] Bourbaki N 1968 Groupes et alg`ebres de Lie (Paris: Hermann) chapitres 4–6 [CY] Cramp´e N and Young C A 2006 Bethe equations for a g2 model J. Phys. A: Math. Gen. 39 L135–43 [Di] van Diejen J F 2004 On the Plancherel formula for the (discrete) Laplacian in aWeyl chamber with repulsive boundary conditions at the walls Ann. Henri Poincar´e 5 135–68 [Do] Dorlas T C 1993 Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schroedinger Model Commun. Math. Phys. 154 347–76 [E] Emsiz E 2006 Affine Weyl groups and integrable systems with delta-potentials PhD Thesis University of Amsterdam, Amsterdam [EOS] Emsiz E, Opdam E M and Stokman J V 2006 Periodic integrable systems with delta-potentials Commun. Math. Phys. 264 191–225 [G1] Gaudin M 1971 Boundary energy of a Bose gas in one dimension Phys. Rev. A 4 386–94 [G2] Gaudin M 1983 La Fonction d’Onde de Bethe (Paris: Masson) [G] Gutkin E 1982 Integrable systems with delta-potential Duke Math. J. 49 1–21 [GS] Gutkin E and Sutherland B 1979 Completely integrable systems and groups generated by reflections Proc. Natl Acad. Sci. USA 76 6057–9 [HO] Heckman G J and Opdam E M 1997 Yang’s system of particles and Hecke algebras Ann. Math. 145 139–73 Heckman G J and Opdam E M 1997 Yang’s system of particles and Hecke algebras Ann. Math. 146 749–50 (erratum) [H] Humphreys J E 1990 Reflection Groups and Coxeter Groups (Cambridge: Cambridge University Press) [K] Korepin V E 1982 Calculations of norms of Bethe wave functions Commun. Math. Phys. 86 391–418 [KBI] Korepin V E, Bogoliubov N M and Izergin A G 1993 Quantum Inverse Scattering Method and Correlation Functions (Cambridge: Cambridge University Press) [LL] Lieb E H and Liniger W 1963 Exact analysis of an interacting Bose gas: I. The general solution and the ground state Phys. Rev. (2) 130 1605–16 [M] Mattis D C (ed) 1994 TheMany-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension (Singapore: World Scientific) [S1] Sutherland B 1980 Nondiffractive scattering: Scattering from kaleidoscopes J. Math. Phys. 21 1770–5 [S2] Sutherland B 2004 Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems (Singapore: World Scientific) [YY] Yang C N and Yang C P 1969 Thermodynamics of a one-dimensional system of Bosons with repulsive delta-function interaction J. Math. Phys. 10 1115–22 13 |

URI: | http://wrap.warwick.ac.uk/id/eprint/30623 |

Data sourced from Thomson Reuters' Web of Knowledge

### Actions (login required)

View Item |