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Journal of Mathematical Physics

The covariant linear oscillator and generalized realization of the dynamical SU(1,1) symmetry algebra

E. D. Kagramanov, R. M. Mir‐Kasimov, Sh. M. Nagiyev

Published Jul 1, 1990• 41 citations
Publisher

Abstract

An exactly solvable problem for the finite-difference Schrödinger equation in the relativistic configurational space is considered. The appropriate finite-difference generalization of the factorization method is developed. The theory of new special functions ‘‘the relativistic Hermite polynomials,’’ in which the solutions are expressed, is constructed.

DOI
10.1063/1.528669

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An exactly solvable problem for the finite-difference Schrödinger equation in the relativistic configurational space is considered. The appropriate finite-difference generalization of the factorization method is developed. The theory of new special functions ‘‘the relativistic Hermite polynomials,’’ in which the solutions are expressed, is constructed.
The concept of the one -- dimensional quantum mechanics in the relativistic configurational space (RQM) is reviewed briefly. The Relativistic Schroedinger equation (RSE) arising here is the finite-difference equation with the step equal to the Compton wave length of the particle. The different generalizations of the Dirac -- Infeld -- Hall factorizarion method for this case are constructed. This method enables us to find out all possible finite-difference generalizations of the most important nonrelativistic integrable case -- the harmonic oscillator. As it was shown (\cite{kmn},\cite{mir6}) in RQM the harmonic oscillator = qqq -- oscillator. It is also shown that the relativistic and nonrelativistic QM's are different representations of the same theory. Thetransformation connecting these two representations is found in explicit form. It could be considered as the generalization of the Kontorovich -- Lebedev transformation.
An exactly solvable relativistic model of a linear oscillator is considered in detail in the presence of a constant external force in both the momentum representation and the relativistic configuration representation. It is found that in contrast to the nonrelativistic case, depending on the magnitude of the force, both discrete and continuous energy spectra are possible. It is shown that in the case of a discrete spectrum, the wave functions in the momentum representation are expressed in terms of the Laguerre polynomials, and in the relativistic configuration representation, in terms of the Meixner–Pollaczek polynomials. Integral and differential–difference formulas are found connecting the Laguerre and Meixner–Pollaczek polynomials. A dynamical symmetry group is constructed.
Exactly solvable NNN-dimensional model of the quantum isotropic singular oscillator in the relativistic configurational r⃗N\vec r_NrN​-space is proposed. It is shown that through the simple substitutions the finite-difference equation for the NNN-dimensional singular oscillator can be reduced to the similar finite-difference equation for the relativistic isotropic three-dimensional singular oscillator. We have found the radial wavefunctions and energy spectrum of the problem and constructed a dynamical symmetry algebra.
Review paper is devoted to the relativistic configuration space (RCS) concept, a version of the relativistic Quantum Mechanics in RCS, the generalization of the Dirac-Infeld-Hall factorization method in the framework of the noncommutative differential calculus natural for RCS, different versions of the deformed oscillators, emerging as the generalization of the harmonic oscillator for RCS. In the formulation of the Newton-Wigner postulates for the relativistic localized states the hypothesis of commutativity of the position operator components is silently accepted as an evident fact. In the present work it is shown that commutativity is not necessary condition and the alternative (noncommutative) approach to the relativistic position operator and localization concept can be realized in a framework of the physically as well as mathematically comprehensive scheme. The different generalizations of the Dirac-Infeld-Hall factorization method for this case are constructed. This method enables us to find out all possible generalizations of the most important nonrelativistic integrable case—the harmonic oscillator. It is shown also that the relativistic oscillator = q —oscillator.
We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann’s functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankel-type integral transform, and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as by-products.
It is shown that the generalization of the quantum harmonic oscillator to the case of the relativistic configurational space is a q-oscillator. The corresponding group of dynamical symmetry is the quantum group SUq(1,1). The deformation parameter being q=e- omega omega 0/ where omega 0=4mc2 and omega is a frequency of the oscillator. The deformed creation and annihilation operators are finite difference ones. The corresponding deformation of the Heisenberg-Weyl group and new coherent states are also considered.