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 SIGMA, 2012, Volume 8, 074, 16 pages (Mi sigma751)

Ladder operators for Lamé spheroconal harmonic polynomials

Ricardo Méndez-Fragosoa, Eugenio Ley-Koob

b Instituto de Física, Universidad Nacional Autónoma de México, México

Abstract: Three sets of ladder operators in spheroconal coordinates and their respective actions on Lamé spheroconal harmonic polynomials are presented in this article. The polynomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lamé polynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers $n_1$ and $n_2$ counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components $\hat L_x$, $\hat L_y$, $\hat L_z$ of the angular momentum connect pairs of the four species of polynomials of a chosen kind and angular momentum. The third set of operators, the cartesian components $\hat p_x$, $\hat p_y$, $\hat p_z$ of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities. Relationships among spheroconal harmonics at the levels of the three sets of operators are illustrated.

Keywords: Lamé polynomials; spheroconal harmonics; ladder operators.

DOI: https://doi.org/10.3842/SIGMA.2012.074

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ArXiv: 1210.4632
MSC: 20C35; 22E70; 33C47; 33C80; 81R05
Received: July 31, 2012; in final form October 9, 2012; Published online October 17, 2012
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Citation: Ricardo Méndez-Fragoso, Eugenio Ley-Koo, “Ladder operators for Lamé spheroconal harmonic polynomials”, SIGMA, 8 (2012), 074, 16 pp.

Citation in format AMSBIB
\Bibitem{MenLey12} \by Ricardo M{\'e}ndez-Fragoso, Eugenio Ley-Koo \paper Ladder operators for Lam\'e spheroconal harmonic polynomials \jour SIGMA \yr 2012 \vol 8 \papernumber 074 \totalpages 16 \mathnet{http://mi.mathnet.ru/sigma751} \crossref{https://doi.org/10.3842/SIGMA.2012.074} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3007285} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000312372800001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84868146599} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Aquilanti V., Marinelli D., Marzuoli A., “Hamiltonian Dynamics of a Quantum of Space: Hidden Symmetries and Spectrum of the Volume Operator, and Discrete Orthogonal Polynomials”, J. Phys. A-Math. Theor., 46:17 (2013), 175303
2. Eugenio Ley-Koo, Guo-Hua Sun, “Surface Effects in the Hydrogen Atom Confined by Dihedral Angles”, Electronic Structure of Quantum Confined Atoms and Molecules, 2014, 1–29
3. Mendez-Fragoso R., Ley-Koo E., “Angular Momentum Theory in Bases of Lame Spheroconal Harmonics”, Concepts of Mathematical Physics in Chemistry: a Tribute To Frank E. Harris - Pt a, Adv. Quantum Chem., 71, eds. Sabin J., CabreraTrujillo R., Elsevier Academic Press Inc, 2015, 115–152
4. Ley-Koo E., “Rotations of the Most Asymmetric Molecules Via 4-Step and 1Step Ladder Operators”, Xxxth international colloquium on group theoretical methods in physics (icgtmp) (group30), Journal of Physics Conference Series, 597, IOP Publishing Ltd, 2015, 012055
5. Medina L., Ley-Koo E., “Family of Lame Spheroconal Quadrupole Harmonic Current Distributions on Spherical Surfaces as Sources of Magnetic Induction Fields With Constant Gradients Inside and Vanishing Asymptotically Outside”, Rev. Mex. Fis., 62:4 (2016), 362–368
6. Sh. Dong, B. J. Falaye, A. E. M. Guerrero, Sh.-H. Dong, “Ladder operators depending on all variables for a charged particle moving in a two-dimensional uniform magnetic field”, J. Adv. Phys., 6:3 (2017), 456–458
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