Riemann surface, 29 addition theorem for sn(u), 28 additive number theory, 15 algebraic function, 29 balanced hypergeometric series, 18 Bell number, 17, 107 Bernoulli counting scheme, 111 Bernoulli number, 17 Bernoulli polynomial, 17 beta integral, 19 binomial series, 18 branched covering map, 28 branchpoint, 29 Catalan number, 16 Chu-Vandermonde summation formula, 19 commutative ordinary differential operators, 117 complete symmetric polynomial, 12 completely multiplicative, 15 complex structure, 29 conformally equivalent, 29 conjugate partition, 5 contiguous relation, 20 Dedekind sum, 107 digamma function, 21 Dirichlet region, 26 double gamma function, 27 Eisenstein series, 26 elementary symmetric polynomial, 12 elliptic function, 26 elliptic integral, 25 Euler number, 22 Euler product, 15 Euler’s dilogarithm, 19 Euler’s transformation formula, 19 Euler-Maclaurin summation formula, 15, 21 Eulerian number, 17, 107 Fermat measure, 53 Fermat’s last theorem, 27 fractional differentiation, 18 function element, 28 fundamental region, 26 Galois field, 65, 121 Gauss second summation theorem, 21 Gauss summation formula, 20 Gegenbauer polynomial, 51 generalised Stirling number, 16 generalised Stirling polynomial, 16 generalized hypergeometric series, 18 generalized Laguerre polynomial, 22 generalized Vandermonde determinant, 125 Genocchi number, 17

The history of q-calculus and a new method

The three degrees of freedom Smorodinsky–Winternitz system is a degenerate or super‐integrable Hamiltonian that possesses five functionally independent globally defined and single‐valued integrals of the motion in both classical and quantum mechanics. This is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the Hamiltonian under a group of symmetry transformations isomorphic to the three‐dimensional unitary unimodular group, SU(3). In turn, this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other, the so‐called dynamical group. All the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group. So, in common with the hydrogen atom and the harmonic oscillator, the quantum mechanics of the Smorodinsky–Winternitz system may be completely solved within the framework of group theory alone.

Group theory of the Smorodinsky-Winternitz system

Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three-dimensional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky-Winternitz potentials and state the corresponding path integral formulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave-functions.

https://cds.cern.ch/record/496404/files/9402121.pdf

Path Integral Discussion for Smorodinsky‐Winternitz Potentials: I. Two‐ and Three Dimensional Euclidean Space

Exactly complete solutions of the Schrödinger equation with a spherically harmonic oscillatory ring-shaped potential

Godel's Proof.

The problem of a particle in the three-dimensional ring-shaped potential ησ2(2a0/r − ηa/r2 sin2 θ)e0 introduced by Hartmann is transformed into the problem of a coupled pair two-dimensional harmonic oscillators with inverse quadratic potentials by using a nonbijective canonical transformation, viz., the Kustaanheimo–Stiefel transformation. The energy E of the levels for the ring-shaped potential is obtained in a straightforward way from the one for the two-dimensional potential — (4Eρ2 + η2σ2a e0/ρ2).

Motion of a particle in a ring-shaped potential: an approach via a nonbijective canonical transformation

The discrete spectrum of a q-analogue of the hydrogen atom is obtained from a deformation of the Pauli equations. As an alternative, the spectrum is derived from a deformation of the four-dimensional oscillator arising in the application of the Kustaanheimo-Stiefel transformation to the hydrogen atom. A model of the 2s-2p Dirac shift is proposed in the context of q-deformations.

https://iopscience.iop.org/article/10.1088/0305-4470/24/22/012/pdf

On the q-analogue of the hydrogen atom

Motion of a particle in a Coulomb plus Aharonov-Bohm potential

The connection between a three-dimensional hydrogenlike atom and a pair of coupled two-dimensional harmonic oscillators is established by applying the Jordan-Schwinger boson calculus to the algebra of the Runge-Lenz-Laplace-Pauli vector. The dynamical groupSO4,2 of the three-dimensional hydrogen atom arises as a quotient of the groupSp8R associated to a four-dimensional harmonic oscillator with constraint.

On the connection between the hydrogen atom and the harmonic oscillator

The connection between the three-dimensional hydrogen atom and a four-dimensional harmonic oscillator obtained in previous works, from a hybridization of the infinitesimal Pauli approach to the hydrogen system with the Schwinger approach to spherical and hyperbolical angular momenta, is worked out in the case of the zero-energy point of the hydrogen atom. This leads to the equivalence of the three-dimensional hydrogen problem with a four-dimensional free-particle problem involving a constraint condition. For completeness, the latter result is also derived by using the Kustaanheimo-Stiefel transformation introduced in celestial mechanics. Finally, it is shown how the Lie algebra of SO(4,2) quite naturally arises for the whole spectrum (discrete plus continuum plus zero-energy point) of the three-dimensional hydrogen atom from the introduction of the constraint condition into the Lie algebra of Sp(8,R) associated with the four-dimensional harmonic oscillator.

Tidjani Négadi

Papers

Motion of a particle in a ring-shaped potential: an approach via a nonbijective canonical transformation

On the q-analogue of the hydrogen atom

Motion of a particle in a Coulomb plus Aharonov-Bohm potential

On the connection between the hydrogen atom and the harmonic oscillator

Connection between the hydrogen atom and the harmonic oscillator: The zero-energy case