Group theoretical aspects of $L^2(\mathbb{R}^+)$, $L^2(\mathbb{R}^2)$ and associated Laguerre polynomials

E. Celeghini; M. A. del Olmo

Group theoretical aspects of $L^2(\mathbb{R}^+)$, $L^2(\mathbb{R}^2)$ and associated Laguerre polynomials

Mathematical Physics 2017-02-08 v1 math.MP

Authors: E. Celeghini , M. A. del Olmo

Abstract

A ladder algebraic structure for $L^2(\mathbb{R}^+)$ which closes the Lie algebra $h(1)\oplus h(1)$ , where $h(1)$ is the Heisenberg-Weyl algebra, is presented in terms of a basis of associated Laguerre polynomials. Using the Schwinger method the quadratic generators that span the alternative Lie algebras $so(3)$ , $so(2,1)$ and $so(3,2)$ are also constructed. These families of (pseudo) orthogonal algebras also allow to obtain unitary irreducible representations in $L^2(\mathbb{R}^2)$ similar to those of the spherical harmonics.

Keywords

group theory associated Laguerre polynomials Lie algebras Heisenberg-Weyl algebra unitary irreducible representations Schwinger method ladder operators

Cite

@article{arxiv.1702.02003,
  title  = {Group theoretical aspects of $L^2(\mathbb{R}^+)$, $L^2(\mathbb{R}^2)$ and associated Laguerre polynomials},
  author = {E. Celeghini and M. A. del Olmo},
  journal= {arXiv preprint arXiv:1702.02003},
  year   = {2017}
}

Comments

6 pages, 31st International Colloquium on Group Theoretical Methods in Physics, Rio de Janeiro, June 19-25, 2016. Accepted in {\sl Springer Proceeding Series}

Group theoretical aspects of $L^2(\mathbb{R}^+)$, $L^2(\mathbb{R}^2)$ and associated Laguerre polynomials

Abstract

Keywords

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