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 twosheeted hyperboloid, where the Lie algebra commutators are the usual PoissonDirac brackets. Then, we introduce a quantized version of the representation in which functions are replaced by operators on a Hilbert space and PoissonDirac 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 nonselfadjoint 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 »
 Authors:

 Departamento de Fisica Teorica, Universidad de Valladolid, 47071 Valladolid (Spain)
 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 00222488
 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; SO4 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 twosheeted hyperboloid, where the Lie algebra commutators are the usual PoissonDirac brackets. Then, we introduce a quantized version of the representation in which functions are replaced by operators on a Hilbert space and PoissonDirac 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 nonselfadjoint 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 twosheeted hyperboloid in terms of the eigenfunctions of the Hamiltonian.},
doi = {10.1063/1.4791683},
journal = {Journal of Mathematical Physics},
issn = {00222488},
number = 2,
volume = 54,
place = {United States},
year = {2013},
month = {2}
}