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SU(2), Associated Laguerre Polynomials and Rigged Hilbert Spaces

  • Enrico Celeghini
  • Manuel Gadella
  • Mariano A. del OlmoEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 255)

Abstract

We present a family of unitary irreducible representations of SU(2) realized in the plane, in terms of the associated Laguerre polynomials. These functions are similar to the spherical harmonics defined on the sphere. Relations with a space of square integrable functions defined on the plane, \(L^2({\mathbb R}^2)\), are analyzed. We have also enlarged this study using rigged Hilbert spaces that allow to work with discrete and continuous bases like is the case here.

Keywords

Lie group representations Special functions Rigged Hilbert spaces 

Notes

Acknowledgements

Partial financial support is acknowledged to the Junta de Castilla y León and FEDER (Project VA057U16) and MINECO of Spain (Project MTM2014-57129-C2-1-P).

References

  1. 1.
    W. Miller Jr. Symmetry and Separation of Variables (Addison-Wesley, Reading MA, 1977).Google Scholar
  2. 2.
    E. Celeghini, M.A. del Olmo, Ann. of Phys. 333 (2013) 90 and 335 (2013) 78.Google Scholar
  3. 3.
    E. Celeghini, M.A. del Olmo, M.A. Velasco, J. Phys.: Conf. Ser. 597 (2015) 012023.Google Scholar
  4. 4.
    E. Celeghini, M.A. del Olmo, in Physical and Mathematical Aspects of Symmetries, ed. by S. Duarte, J.P. Gazeau, et al (Springer, New York, 2017).Google Scholar
  5. 5.
    E. Celeghini, M.A. del Olmo, J. Phys.: Conf. Ser. 597 (2015) 012022.Google Scholar
  6. 6.
    E. Celeghini, M. Gadella, M.A. del Olmo, J. Math. Phys. 57 (2016) 072105.MathSciNetCrossRefGoogle Scholar
  7. 7.
    E. Celeghini, M. Gadella, M.A. del Olmo, J. Math. Phys. 59 (2018) 053502.Google Scholar
  8. 8.
    G. Lindblad, B Nagel, Ann. Inst. Henry Poincaré 13 (1970) 27–56.Google Scholar
  9. 9.
    A. Böhm, The Rigged Hilbert Space and Quantum Mechanics, Springer Lecture Notes in Physics 78 (Springer, Berlin, 1978).CrossRefGoogle Scholar
  10. 10.
    G. Szegö, Orthogonal Polynomials (Am. Math. Soc., Providence, 2003).zbMATHGoogle Scholar
  11. 11.
    F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (edit.) NIST Handbook of Mathematical Functions (Cambridge Univ. Press, New York, 2010).Google Scholar
  12. 12.
    Wu-Ki Tung, Group Theory in Physics (World Sc., Singapore, 1985).CrossRefGoogle Scholar
  13. 13.
    I.M. Gelf’and, N.Ya. Vilenkin, Generalized Functions: Applications to Harmonic Analysis (Academic, New York, 1964).Google Scholar
  14. 14.
    A. Bohm, M. Gadella, Dirac Kets, Gamow vectors and Gelfand Triplets, Springer Lecture Notes in Physics, 348 (Springer, Berlin, 1989).CrossRefGoogle Scholar
  15. 15.
    J. Horváth, Topological Vector Spaces and Distributions (Addison-Wesley, Reading MA., 1966).zbMATHGoogle Scholar
  16. 16.
    A. Pietsch, Nuclear Topological Vector Spaces (Springer, Berlin, 1972).zbMATHGoogle Scholar
  17. 17.
    M. Gadella, F. Gómez, Found. Phys., 32 (2002) 815–869.MathSciNetCrossRefGoogle Scholar
  18. 18.
    M. Reed, B. Simon, Functional Analysis (Academic, New York, 1972).zbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Enrico Celeghini
    • 1
    • 2
  • Manuel Gadella
    • 3
  • Mariano A. del Olmo
    • 3
    Email author
  1. 1.Dpto di FisicaUniversità di Firenze and INFN–Sezione di FirenzeFirenzeItaly
  2. 2.Dpto. de Física TeóricaUniversidad de ValladolidValladolidSpain
  3. 3.Dpto de Física Teórica and IMUVAUniv. de ValladolidValladolidSpain

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