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Heisenberg algebra, umbral calculus and orthogonal polynomials

J. Math. Phys. 49, 053509 (2008); doi:10.1063/1.2909731

Published 7 May 2008

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G. Dattoli,1 D. Levi,2 and P. Winternitz3
1ENEA, Dipartimento Fim, Cre Frascati, C.P. 65, Frascati, Rome 000044, Italy
2Dipartimento di Ingegneria Elettronica, Universitá Degli Studi Roma Tre and Sezione INFN Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
3Centre de Recherches Mathématiques and Department de Mathématiques et de Statistiques, Université de Montréal, C.P. 6128 Centre Ville, Montréal, Quebec H3C 3J7, Canada
Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation [P-hat,M-hat]=1. In ordinary quantum mechanics, P-hat is the derivative and M-hat the coordinate operator. Here, we shall realize P-hat as a second order differential operator and M-hat as a first order integral one. We show that this makes it possible to solve large classes of differential and integrodifferential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials are particularly well suited for describing the so-called flatenned beams in laser theory ©2008 American Institute of Physics
History: Received 3 December 2007; accepted 26 February 2008; published 7 May 2008
Permalink: http://link.aip.org/link/?JMAPAQ/49/053509/1
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KEYWORDS and PACS

Keywords
PACS
  • 42.55.Ah
    General laser theory
  • 03.65.Db
    Functional analytical methods in quantum mechanics
  • 03.65.Fd
    Algebraic methods in quantum mechanics
  • 02.30.Sa
    Functional analysis
  • 02.10.-v
    Logic, set theory, and algebra
  • YEAR: 2008

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ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (26)
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