Shape Invariant Rational Extensions And Potentials Related to Exceptional Polynomials

Abstract

In this paper, we show that an attempt to construct shape invariant extensions of a known shape invariant potential leads to, apart from a shift by a constant, the well known technique of isospectral shift deformation. Using this, we construct infinite sets of generalized potentials with $X_m$ exceptional polynomials as solutions. These potentials are rational extensions of the existing shape invariant potentials. The method is elucidated using the radial oscillator and the trigonometric Pöschl-Teller potentials. For the case of radial oscillator, in addition to the known rational extensions, we construct two infinite sets of rational extensions, which seem to be less studied. For one of the potential, we show that its solutions involve a third type of exceptional Laguerre polynomials. Explicit expressions of this generalized infinite set of potentials and the corresponding solutions are presented. For the trigonometric Pöschl-Teller potential, our analysis points to the possibility of several rational extensions beyond those known in literature.

18 pages, 1 figure

References in corpus (6)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Solvable rational extensions of the isotonic oscillator
• Exceptional orthogonal polynomials and the Darboux transformation
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Cited by in corpus (6)

• Generation of a Novel Exactly Solvable Potential
• The Supersymmetric WKB Formalism is Not Exact for All Additive Shape Invariant Potentials
• Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials
• Generalized coherent states of exceptional Scarf-I potential: Their spatio-temporal and statistical properties
• Construction of 2nd stage shape invariant potentials
• QHJ route to multi-indexed exceptional Laguerre polynomials and corresponding rational potentials