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Rational extensions of the trigonometric Darboux-Poschl-Teller potential based on para-Jacobi polynomials

Abstract : The possibility for the Jacobi equation to admit, in some cases, general solutions that are polynomials has been recently highlighted by Calogero and Yi, who termed them para-Jacobi polynomials. Such polynomials are used here to build seed functions of a Darboux-Backlund transformation for the trigonometric Darboux-Poschl-Teller potential. As a result, one-step regular rational extensions of the latter depending both on an integer index n and on a continuously varying parameter lambda are constructed. For each n value, the eigenstates of these extended potentials are associated with a novel family of lambda-dependent polynomials, which are orthogonal on]-1,1[.
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B. Bagchi, Y. Grandati, C. Quesne. Rational extensions of the trigonometric Darboux-Poschl-Teller potential based on para-Jacobi polynomials. Journal of Mathematical Physics, American Institute of Physics (AIP), 2015, 56 (6), ⟨10.1063/1.4922017⟩. ⟨hal-01513176⟩

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