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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 and the SGA is . We start with a representation of 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.

© 2013 American Institute of Physics

Received 04 April 2012
Accepted 28 January 2013
Published online 19 February 2013

Acknowledgments:
In part financial support is indebt to the Ministry of Science of Spain (Project Nos. MTM2009-10751 and FIS2009-09002), and to the Russian Science Foundation (Grant No. 10-01-00300).

Article outline:

I. INTRODUCTION
II. A CLASSICAL PARTICLE IN A THREE-DIMENSIONAL HYPERBOLOID
III. THE QUANTUM CASE
A. Restrictive relations
IV. THE QUANTUM FREE HAMILTONIAN ON
A. The ladder operators
V. GENERAL PROPERTIES OF THE EIGENFUNCTIONS OF THE HAMILTONIAN
VI. CONCLUDING REMARKS AND DISCUSSION

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2013-02-19

2016-10-24

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