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

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-Bäcklund transformation for the trigonometric Darboux-Pöschl-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 $λ$ are constructed. For each $n$ value, the eigenstates of these extended potentials are associated with a novel family of $λ$-dependent polynomials, which are orthogonal on $\left] -1,1\right[$.

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