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Title: Spectrum generating algebra for the continuous spectrum of a free particle in Lobachevski space

Abstract

In this paper, we construct a Spectrum Generating Algebra (SGA) for a quantum system with purely continuous spectrum: the quantum free particle in a Lobachevski space with constant negative curvature. The SGA contains the geometrical symmetry algebra of the system plus a subalgebra of operators that give the spectrum of the system and connects the eigenfunctions of the Hamiltonian among themselves. In our case, the geometrical symmetry algebra is so(3,1) and the SGA is so(4,2). We start with a representation of so(4,2) by functions on a realization of the Lobachevski space given by a two-sheeted hyperboloid, where the Lie algebra commutators are the usual Poisson-Dirac brackets. Then, we introduce a quantized version of the representation in which functions are replaced by operators on a Hilbert space and Poisson-Dirac brackets by commutators. Eigenfunctions of the Hamiltonian are given and 'naive' ladder operators are identified. The previously defined 'naive' ladder operators shift the eigenvalues by a complex number so that an alternative approach is necessary. This is obtained by a non-self-adjoint function of a linear combination of the ladder operators, which gives the correct relation among the eigenfunctions of the Hamiltonian. We give an eigenfunction expansion of functions over the upper sheetmore » of a two-sheeted hyperboloid in terms of the eigenfunctions of the Hamiltonian.« less

Authors:
; ;  [1];  [2]
  1. Departamento de Fisica Teorica, Universidad de Valladolid, 47071 Valladolid (Spain)
  2. Department of Theoretical Physics, IHEP, Protvino, Moscow Region 142280 (Russian Federation)
Publication Date:
OSTI Identifier:
22162768
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 54; Journal Issue: 2; Other Information: (c) 2013 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; EIGENFUNCTIONS; EIGENVALUES; EXPANSION; HAMILTONIANS; HILBERT SPACE; LOBACHEVSKY GEOMETRY; PARTICLES; QUANTIZATION; SO-4 GROUPS; SPECTRA

Citation Formats

Gadella, M., Negro, J., Santander, M., and Pronko, G. P. Spectrum generating algebra for the continuous spectrum of a free particle in Lobachevski space. United States: N. p., 2013. Web. doi:10.1063/1.4791683.
Gadella, M., Negro, J., Santander, M., & Pronko, G. P. Spectrum generating algebra for the continuous spectrum of a free particle in Lobachevski space. United States. doi:10.1063/1.4791683.
Gadella, M., Negro, J., Santander, M., and Pronko, G. P. Fri . "Spectrum generating algebra for the continuous spectrum of a free particle in Lobachevski space". United States. doi:10.1063/1.4791683.
@article{osti_22162768,
title = {Spectrum generating algebra for the continuous spectrum of a free particle in Lobachevski space},
author = {Gadella, M. and Negro, J. and Santander, M. and Pronko, G. P.},
abstractNote = {In this paper, we construct a Spectrum Generating Algebra (SGA) for a quantum system with purely continuous spectrum: the quantum free particle in a Lobachevski space with constant negative curvature. The SGA contains the geometrical symmetry algebra of the system plus a subalgebra of operators that give the spectrum of the system and connects the eigenfunctions of the Hamiltonian among themselves. In our case, the geometrical symmetry algebra is so(3,1) and the SGA is so(4,2). We start with a representation of so(4,2) by functions on a realization of the Lobachevski space given by a two-sheeted hyperboloid, where the Lie algebra commutators are the usual Poisson-Dirac brackets. Then, we introduce a quantized version of the representation in which functions are replaced by operators on a Hilbert space and Poisson-Dirac brackets by commutators. Eigenfunctions of the Hamiltonian are given and 'naive' ladder operators are identified. The previously defined 'naive' ladder operators shift the eigenvalues by a complex number so that an alternative approach is necessary. This is obtained by a non-self-adjoint function of a linear combination of the ladder operators, which gives the correct relation among the eigenfunctions of the Hamiltonian. We give an eigenfunction expansion of functions over the upper sheet of a two-sheeted hyperboloid in terms of the eigenfunctions of the Hamiltonian.},
doi = {10.1063/1.4791683},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 2,
volume = 54,
place = {United States},
year = {2013},
month = {2}
}