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The Factorization Method, Self-Similar Potentials and Quantum Algebras

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Special Functions 2000: Current Perspective and Future Directions

Part of the book series: NATO Science Series ((NAII,volume 30))

Abstract

The factorization method is a convenient operator language formalism for consideration of certain spectral problems. In the simplest differential operators realization it corresponds to the Darboux transformations technique for linear ODE of the second order. In this particular case the method was developed by Schrödinger and became well known to physicists due to the connections with quantum mechanics and supersymmetry. In the theory of orthogonal polynomials its origins go back to the Christoffel’s theory of kernel polynomials, etc. Special functions are defined in this formalism as the functions associated with similarity reductions of the factorization chains.

We consider in this lecture in detail the Schrödinger equation case and review some recent developments in this field. In particular, a class of selfsimilar potentials is described whose discrete spectrum consists of a finite number of geometric progressions. Such spectra are generated by particular polynomial quantum algebras which include q-analogues of the harmonic oscillator and su(1, 1) algebras. Coherent states of these potentials are described by differential-delay equations of the pantograph type. Applications to infinite soliton systems, Ising chains, random matrices, and lattice Coulomb gases are briefly outlined.

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Author information

Authors and Affiliations

  1. Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow region, 141980, Russia

    V. P. Spiridonov

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  1. V. P. Spiridonov
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Editors and Affiliations

  1. Department of Mathematics, Arizona State University, Tempe, Arizona, USA

    Joaquin Bustoz  & Sergei K. Suslov  & 

  2. Department of Mathematics, University of South Florida, Tampa, Florida, USA

    Mourad E. H. Ismail

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Spiridonov, V.P. (2001). The Factorization Method, Self-Similar Potentials and Quantum Algebras. In: Bustoz, J., Ismail, M.E.H., Suslov, S.K. (eds) Special Functions 2000: Current Perspective and Future Directions. NATO Science Series, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0818-1_13

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  • DOI: https://doi.org/10.1007/978-94-010-0818-1_13

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-7120-5

  • Online ISBN: 978-94-010-0818-1

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