Spheroidal analysis of the generalized MIC-Kepler system

This paper deals with the dynamical system that generalizes the MIC-Kepler system. It is shown that the Schrödinger equation for this generalized MIC-Kepler system can be separated in prolate spheroidal coordinates. The coefficients of the interbasis expansions between three bases (spherical, parabolic, and spheroidal) are studied in detail. It is found that the coefficients for this expansion of the parabolic basis in terms of the spherical basis, and vice versa, can be expressed through the Clebsch-Gordan coefficients for the group SU(2) analytically continued to real values of their arguments. The coefficients for the expansions of the prolate spheroidal basis in terms of the spherical and parabolic bases are proved to satisfy three-term recursion relations.     

Four-dimensional double-singular oscillator

The Schrödinger equation for the four-dimensional double-singular oscillator is separable in Eulerian, double-polar, and spheroidal coordinates in ?⁴. It is shown that the coefficients for the expansion of the double-polar basis in terms of the Eulerian basis can be expressed through the Klebsch-Gordan coefficients of the group SU(2) analytically continued to real values of their arguments. The coefficients for the expansions of the spheroidal basis in terms of the Eulerian and double-polar bases are proved to satisfy three-term recursion relations.     

Feynman operator calculus and singular quantum oscillator

New applications of Feynman disentangling method in quantum mechanics are studied and the time-dependent singular oscillator problem is solved in this approach. The important role of representation group theory is discussed in this context.     

Canonical symmetry properties of the constrained singular generalized mechanical system

Based on generalized Apell－Chetaev constraint conditions and to take the inherent constrains for singular Lagrangian into account, the generalized canonical equations for a general mechanical system with a singular higher-order Lagrangian and subsidiary constrains are formulated. The canonical symmetries in phase space for such a system are studied and Noether theorem and its inversion theorem in the generalized canonical formalism have been established.     

Theory of the singular quantum oscillator

We propose a number of arguments in favor of reevaluating the theory of a quantum oscillator described by the Hamiltonian H=–d²/dx² + 2²x² + x^–2(=2m=1). We propose that functions ₊(x) which continuously reduce to even harmonic oscillator solutions in the 0 limit be taken as the even solutions of the Hamiltonian in the –1/4 < < 3/4 range. In this scheme the problem becomes truly one-dimensional such that even and odd parity energy levels alternate, whereas the usual approach leads to parity degeneracy.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 85–89, October, 1989.     

Representations of generalized oscillator algebra

The representations of the oscillator algebra introduced by Brzeziski et al. [Phys. Lett. B 311 (1993) 202] are classified.     

Feynman method for disentangling operators and a singular oscillator

The problem of the evolution of a singular quantum oscillator with a frequency exhibiting an arbitrary time dependence has been solved. The probabilities w _mn of transitions in the oscillator spectrum and generating functions have been calculated, and the sum rules for w _mn have been derived. The calculations are based on the Feynman disentangling method and the theory of representations of the SU(1, 1) group.     

Representation theory of generalized deformed oscillator algebras

The representation theory of the generalized deformed oscillator algebras (GDOA's) is developed. GDOA's are generated by the four operators {1, a, a , N}. Their commutators and Hermiticity properties are those of the boson oscillator algebra, except for [a, a ] _q = G(N), where [a, b] _q = ab – q ba and G(N) is a Hermitian, analytic function. The unitary irreductible representations are obtained by means of a Casimir operator C and the semi-positive operator a a. They may belong to one out of four classes: bounded from below (BFB), bounded from above (BFA), finite-dimentional (FD), unbounded (UB). Some examples of these different types of unirreps are given.     

Quantum phases for a generalized harmonic oscillator

An effective Hamiltonian for the generalized harmonic oscillator is determined by using squeezed state wavefunctions. The equations of motion over an extended phase space are determined and then solved perturbatively for a specific choice of the oscillator parameters. These results are used to calculate the dynamic and geometric phases for the generalized oscillator with this choice of parameters.      

The effect of singular potentials on the harmonic oscillator

We address the problem of a quantum particle moving under interactions presenting singularities. The self-adjoint extension approach is used to guarantee that the Hamiltonian is self-adjoint and to fix the choice of boundary conditions. We specifically look at the harmonic oscillator added of either a δ-function potential or a Coulomb potential (which is singular at the origin). The results are applied to Landau levels in the presence of a topological defect, the Calogero model and to the quantum motion on the noncommutative plane.     

Models for the 3D singular isotropic oscillator quadratic algebra

We give the first explicit construction of the quadratic algebra for a 3D quantum superintegrable system with nondegenerate (4-parameter) potential together with realizations of irreducible representations of the quadratic algebra in terms of differential—differential or differential—difference and difference—difference operators in two variables. The example is the singular isotropic oscillator. We point out that the quantum models arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras for superintegrable systems in n dimensions and are closely related to Hecke algebras and multivariable orthogonal polynomials.     

Holomorphic representation of coherent states of a singular oscillator and their Darboux representation

We consider the coherent states of a singular oscillator; these states are defined to be the characteristic states of an operator that reduces the number of basis functions for a discrete spectrum to one. We use the Darboux transform to study the coherent states of a transformed Hamiltonian. We obtain an expression for measures that can be used to decompose unity. We construct a holomorphic representation for the state vectors in the space of functions holomorphic everywhere in the complex plane, including vectors for discrete spectra and coherent states. We obtain a holomorphic representation of the Darboux transformation operators. Tomsk State University. Institute of High-Energy Electronics. Siberian Division of the Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 46–53, February, 1998.     

Bohlin transformation for a two-dimensional singular oscillator in a constant magnetic field

Bohlin transformation for a circular singular oscillator in a constant magnetic field is considered. It is shown that this transformation leads to a two-dimensional Kepler problem with an additional centrifugal potential from the constant magnetic field whose strength decreases inversely proportional to the distance from the center of attraction of the system. The energy spectrum of the considered system is obtained.     

Specific heat of the harmonic oscillator within generalized equilibrium statistics

Summary Within the generalized equilibrium statistics recently introduced by Tsallis (p _n ∝[1−β(q−-1) _εn ]^1/(q−)), we calculate the thermal dependence of the specific heat corresponding to a harmonic-oscillator-like spectrum, namely ε _n ω(n−α) (∀ω>0,n=0,1,2,...). The influences ofq and α are exhibited. Physically inaccessible and/or thermally frozen gaps are obtained in the low-temperature region, and, forq>1, oscillations are observed in the high-temperature region. The specific heat of the two-level system is also shown.     

On peculiarities of generalized oscillator strengths for ionization from p-states

Exact expressions are given in the Born approximation for the density of the generalized oscillator strenghts of hydrogen-like particles for electron transitions from the states 3p₀ and 3p_±1 into the continuum with zero energy. The peculiarities of the generalized oscillator strengths are considered.     

Certain class of generalized isotropic perturbations of thes-wave three-dimensional harmonic oscillator

In this paper we study a certain class of generalized perturbation problem for the isotropic three dimensional harmonic oscillator, forl=0 states only, for all excited states. The efficacy of our formalism is tested by studying the Killingbeck potential in detail. We also briefly present our results using the diagonal Padé approximants.     

On the quantum motion of a generalized time-dependent forced harmonic oscillator

In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the Schrödinger equation for the generalized time-dependent forced harmonic oscillator in terms of solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. In addition, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum as well as the quantum correlations between coordinate and momentum. It is shown that the width of the Gaussian packet, fluctuations and correlations do not depend on the external force. As a particular case, we consider the forced Caldirola-Kanai oscillator.     

The Nambu mechanics as a class of singular generalized dynamical formalisms

In a recent work Nambu has proposed ac-number dynamical formalism which can allow an odd numbern of canonical variables. Naturally associated to this new mechanics there exists ann-linear bracket whose study opens interesting possibilities. The purpose of this work is to show that besides this bracket another one which is bilinear and in fact a Lie bracket can also be associated with the Nambu mechanics. For anyn, however, this bracket is singular. In a sense previously used by the present author, this result exhibits the Nambu mechanics as an interesting class of singular generalized dynamical formalisms irrespective of the number of phase space variables. Reasons are given suggesting that such singular formalisms would be, within our context, the only ones capable of describing classical analogues of generalized quantum systems.