Abstract:
A study is made of the problem of diagonal operators on a two-dimensional sphere. A trigonometrie
form of an elliptic system of coordinates on a sphere that is convenient for applications
in physics is derived. Wave eigenfunctions of diagonal operators in the elliptic coordinate
system – so-called spheroconieal functions – are constructed. Their main properties
are derived. Conditions that determine the eigenvalues of the second diagonal operator
in the elliptic coordinate system are found. Some matrix elements of spheroconicM functions
are calculated. Possible applications in physics are discussed for the complete set of
quantum-mechanical observables associated with the elliptic coordinate system on the twodimensional
sphere.
Citation:
I. Lukach, “A complete set of quantum-mechanical observables on a two-dimensional sphere”, TMF, 14:3 (1973), 366–380; Theoret. and Math. Phys., 14:3 (1973), 271–281
\Bibitem{Luk73}
\by I.~Lukach
\paper A complete set of quantum-mechanical observables on a two-dimensional sphere
\jour TMF
\yr 1973
\vol 14
\issue 3
\pages 366--380
\mathnet{http://mi.mathnet.ru/tmf3395}
\transl
\jour Theoret. and Math. Phys.
\yr 1973
\vol 14
\issue 3
\pages 271--281
\crossref{https://doi.org/10.1007/BF01029309}
Linking options:
https://www.mathnet.ru/eng/tmf3395
https://www.mathnet.ru/eng/tmf/v14/i3/p366
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Pogosyan G.S., Yakhno A., “Separations of Variables and Analytic Contractions on Two-Dimensional Hyperboloids”, Phys. Part. Nuclei, 50:2 (2019), 87–140
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J. Freimanis, “Polarized radiative transfer equation in some curvilinear coordinate systems”, Journal of Quantitative Spectroscopy and Radiative Transfer, 146 (2014), 250
Ricardo Méndez-Fragoso, Eugenio Ley-Koo, “Ladder operators for Lamé spheroconal harmonic polynomials”, SIGMA, 8 (2012), 074, 16 pp.
Pogosyan, GS, “Lie-algebra contractions and separation of variables. Three-dimensional sphere”, Physics of Atomic Nuclei, 72:5 (2009), 836
Pogosyan, G, “Separation of variables and Lie algebra contractions. Applications to special functions”, Physics of Particles and Nuclei, 33 (2002), S123
Grosche, C, “Handbook of Feynman path integrals - Introduction”, Handbook of Feynman Path Integrals, 145 (1998), 1
C Grosche, Kh H Karayan, G S Pogosyan, A N Sissakian, “Quantum motion on the three-dimensional sphere: the ellipso-cylindrical bases”, J. Phys. A: Math. Gen., 30:5 (1997), 1629
A A Izmest'ev, G S Pogosyan, A N Sissakian, P Winternitz, “Contractions of Lie algebras and separation of variables”, J. Phys. A: Math. Gen., 29:18 (1996), 5949
C. Grosche, G. S. Pogosyan, A. N. Sissakian, “Path Integral Discussion for Smorodinsky-Winternitz Potentials: II. The Two- and Three-Dimensional Sphere”, Fortschr. Phys., 43:6 (1995), 523
I. V. Komarov, “Kowalewski basis for the hydrogen atom”, Theoret. and Math. Phys., 47:1 (1981), 320–324
I. Lukach, “Complete sets of observables on the sphere in four-dimensional Euclidean space”, Theoret. and Math. Phys., 31:2 (1977), 457–461
J. Patera, P. Winternitz, “On bases for irreducible representations of O (3) suitable for systems with an arbitrary finite symmetry group”, The Journal of Chemical Physics, 65:7 (1976), 2725
D. I. Abramov, I. V. Komarov, “Phase-shift method for scattering on potentials that allow separation of variables in spheroidal coordinates”, Theoret. and Math. Phys., 22:2 (1975), 179–183
I. Lukach, Ya. A. Smorodinskii, “Separation of variables in a spheroconical coordinate system and the Schrödinger equation for a case of noncentral forces”, Theoret. and Math. Phys., 14:2 (1973), 125–131
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