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 [Formula: see text] exceptional polynomials as solutions. The method is simple and transparent and 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. Explicit expressions of the 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.