$m$-step rational extensions of the trigonometric Darboux-Pöschl-Teller potential based on para-Jacobi polynomials

TL;DR

The paper addresses constructing regular rational extensions of the trigonometric Darboux-Pöschl-Teller potential via $m$-step Darboux transformations using para-Jacobi seed functions. It extends previous one-step results to multi-step chains, deriving extended potentials and eigenfunctions in terms of Wronskian/determinant structures and introducing novel families of exceptional orthogonal polynomials depending on $m$ discrete indices and $m$ continuous parameters $oldsymbol{ extλ}$. The authors establish detailed regularity conditions for the continuous parameters and seed-index sequences, provide explicit formulae for the extended potentials and spectra, and illustrate with concrete $(N,M)=(3,3)$ examples. This work broadens the landscape of EOPs by linking multi-step DT constructions to a rich family of $ extit{λ}$-dependent EOPs with potential applications in exactly solvable quantum systems and related areas.

Abstract

A previous construction of regular rational extensions of the trigonometric Darboux-Pöschl-Teller potential, obtained by one-step Darboux transformations using seed functions associated with the para-Jacobi polynomials of Calogero and Yi, is generalized by considering $m$-step Darboux transformations. As a result, some novel families of exceptional orthogonal polynomials depending on $m$ discrete parameters, as well as $m$ continuous real ones $λ_1$, $λ_2$, \ldots, $λ_m$, are obtained. The restrictions imposed on these parameters by the rational extensions regularity conditions are studied in detail.

Figures (3)

• Figure 1: Potential $V^{(-5,-4)}(z;3,3;-1,1)$ in terms of $z$ for $-1<z<1$.
• Figure 2: Potential $V^{(-5,-4)}(z;3,3;t,1)$ in terms of $z$ and $t$ for $-1<z<1$ and $-2<t<0$.
• Figure 3: Potential $V^{(-5,-4)}(z;3,3;-1,\mu)$ in terms of $z$ and $\mu$ for $-1<z<1$ and $0<\mu<3$.