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

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.

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