Exactly solvable potentials and Romanovski polynomials in quantum mechanics.

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Exactly solvable potentials and Romanovski
polynomials in quantum mechanics.

• David Edwin Álvarez Castillo
• July 16, 2008

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Classical Orthogonal Polynomials

• Legendre
• Laguerre
• Hermite
• Chebyshev
• Gegenbauer
• Jacobi

Edmond Nicolas Laguerre 1834 - 1886
Adrien Marie Legendre 1752 - 1833
Charles Hermite 1822-1901
Pafnuty Lvovich Chebyshev 1821 - 1894
Carl Gustav Jacobi Jacobi 1804 - 1851
Leopold Bernhard Gegenbauer 1849 - 1903
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Exactly solvable potentials in quantum mechanics
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Hyperbolic Scarf Potential
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Solution
Self-adjoint form
in terms of the Rodrigues formula
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Real solutions in terms of Romanovski polynomials
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4.3 Romanovski Polynomials and the non spherical
angular functions
Consider an electron in the following potential
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The angular equation from the SE has the solution
The total wave function is
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Relation between the associated Legendre
functions and Romanovski polynomials if c0
(central potential)
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Spherical Harmonics VS non spherical angular
functions
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Romanovski polynomials in the trigonometric
Rosen-Morse
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A taylor expansion shows

• First term Coulomb.
• Second term linear confinement.
• Third term standard centrifugal barrier.

In this sense, Rosen-Morse I can be viewed as
the image of space-like gluon propagation in
coordinate space. Compean, Kirchbach (2006).
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• Advantages of the RMt over the Coulomb potential
  lineal (QCD)
• Dynamical symmetry O(4),
• Exact solutions,
• Good description of nucleons spectrum.

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The nucleon excitation spectrum below 2
GeV. (Courtesy M. Kirchbach)
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Summary

• The Romanovski polynomials appear as the solution
  of the Schrodinger equation for the Hyperbolic
  Scarf Potential and the Rosen-Morse
  trigonometric.
• They define new non-spherical angular functions.
• The Romanovski polynomials are the main designers
  of non--spherical angular functions of a new
  type, which we identified with components of the
  eigenvectors of the infinite discrete unitary
  SU(1,1) representation,

References quant-ph/0603122 arXiv0706.3897
quant-ph/0603232
GRACIAS