Coherent orthogonal polynomials
We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these onedimensional continuous spaces allow, besides the standard uncountable basis (x〉), for an alternative countable basis (n〉). The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the halfline and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinitedimensional irreducible representation of a noncompact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extendedmore »
 Authors:

^{[1]};
^{[2]}
 Dipartimento di Fisica, Università di Firenze and INFN–Sezione di Firenze, I50019 Sesto Fiorentino, Firenze (Italy)
 Departamento de Física Teórica and IMUVA, Universidad de Valladolid, E47005, Valladolid (Spain)
 Publication Date:
 OSTI Identifier:
 22220766
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Annals of Physics (New York); Journal Volume: 335; Other Information: Copyright (c) 2013 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; ANNIHILATION OPERATORS; DIFFERENTIAL EQUATIONS; EIGENSTATES; HERMITE POLYNOMIALS; HILBERT SPACE; INTEGRAL CALCULUS; LEGENDRE POLYNOMIALS; LIE GROUPS; MATRIX ELEMENTS; QUANTUM MECHANICS; RECURSION RELATIONS