Abstract
The spherical and parabolic wave functions are calculated for the generalized MIC–Kepler system in the continuous spectrum. It is shown that the coefficients of the parabola–sphere and sphere–parabola expansion are expressed in terms of the generalized hypergeometric function \(_{3}F_2(\ldots\mid 1)\). The quantum mechanical problem of scattering in the generalized MIC–Kepler system is solved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Mardoyan, “The generalized MIC–Kepler system,” J. Math. Phys., 44, 4981–4987 (2003).
D. Zwanziger, “Exactly soluble nonrelativistic model of particles with both electric and magnetic charges,” Phys. Rev., 176, 1480–1488 (1968).
H. McIntosh and A. Cisneros, “Degeneracy in the presence of a magnetic monopole,” J. Math. Phys., 11, 896–916 (1970).
P. Kustaanheimo, A. Schinzel, H. Davenport, and E. Stiefel, “Perturbation theory of Kepler motion based on spinor regularization,” J. Reine Angew. Math., 1965, 204–219 (1965).
T. Iwai and Y. Uwano, “The quantised MIC–Kepler problem and its symmetry group for negative energies,” J. Phys. A: Math. Gen., 21, 4083–4104 (1988).
A. Nersessian and V. Ter-Antonyan, “ ‘Charge-dyon’ system as the reduced oscillator,” Modern Phys. Lett. A, 9, 2431–2435 (1994).
V. Ter-Antonyan and A. Nersessian, “Quantum oscillator and a bound system of two dyons,” Modern Phys. Lett. A, 10, 2633–2638 (1995).
A. Nersessian, V. Ter-Antonyan, and M. M. Tsulaia, “A note on quantum Bohlin transformation,” Modern Phys. Lett. A, 11, 1605–1610 (1996).
A. P. Nersessian and V. M. Ter-Antonyan, “Anyons, monopoles, and Coulomb problem,” arXiv: physics/9712027.
M. Kibler, L. G. Mardoyan, and G. S. Pogosyan, “On a generalized Kepler–Coulomb system: Interbasis expansions,” Int. J. Quantum Chem., 52, 1301 (1994).
J. Friš, V. Mandrosov, Ya. A. Smorodinsky, M. Uhíř, and P. Winternitz, “On higher symmetries in quantum mechanics,” Phys. Lett., 16, 354–356 (1965).
N. W. Evans, “Super-integrability of the Winternitz system,” Phys. Lett. A, 147, 483–486 (1990).
H. Hartmann, “Die Bewegung eines Körpers in einem ringförmigen Potentialfeld,” Theor. Chim. Acta, 24, 201–206 (1972); H. Hartmann, R. Schuch, and J. Radke, “Die diamagnetische Suszeptibilität eines nicht kugelsymmetrischen Systems,” Theor. Chim. Acta, 42, 1–3 (1976); H. Hartmann and R. Schuch, “Spin-orbit coupling for the motion of a particle in a ring-shaped potential,” Int. J. Quantum Chem., 18, 125–141 (1980); C. Quesne, “A new ring-shaped potential and its dynamical invariance algebra,” J. Phys. A: Math. Gen., 21, 3093–3101 (1988).
L. G. Mardoyan, “Spheroidal analysis of the generalized MIC–Kepler system,” Phys. Atom. Nucl., 68, 1746–1755 (2005).
L. G. Mardoyan and M. G. Petrosyan, “Four-dimensional singular oscillator and generalized MIC–Kepler system,” Phys. Atom. Nucl., 70, 572–575 (2007).
I. Marquette, “Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras,” J. Math. Phys., 51, 102105, 10 pp. (2010); H. Shmavonyan, “\(\mathbb{C}^N\)-Smorodinsky–Winternitz system in a constant magnetic field,” Phys. Lett. A, 383, 1223–1228 (2019).
L. G. Mardoyan, “Scattering of Electrons on the Dyon,” Theoret. and Math. Phys., 136, 1110–1118 (2003).
I. E. Tamm, Selected Papers (B. M. Bolotovskii, V. Ya. Frenkel, and R. Peierls), Springer, Berlin (1991).
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York–Toronto–London (1953).
L. G. Mardoyan, “Ring-Shaped Functions and Wigner \(6j\)-Symbols,” Theoret. and Math. Phys., 146, 248–258 (2006).
L. D. Landau and E. M. Lifshitz, Course of theoretical physics, Vol. 3: Quantum mechanics: non-relativistic theory (Addison-Wesley Series in Advanced Physics), Pergamon Press Ltd., London–Paris (1958).
G. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1: The Hypergeometric Function. Legendre Functions, McGraw-Hill, New York (1953).
L. G. Mardoyan, G. S. Pogosyan, A. N. Sisakyan, and V. M. Ter-Antonyan, Quantum Systems with Hidden Symmetry [in Russian], Fizmatlit, Moscow (2006).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflicts of interest.
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 285–298 https://doi.org/10.4213/tmf10543.
Rights and permissions
About this article
Cite this article
Mardoyan, L.G. Bases and interbasis expansions in the generalized MIC–Kepler problem in the continuous spectrum and the scattering problem. Theor Math Phys 217, 1661–1672 (2023). https://doi.org/10.1134/S004057792311003X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S004057792311003X