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 Xm exceptional polynomials as solutions. The method is simple and transparent and is elucidated using the radial oscillator and the trigonometric PöschlTeller 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 PöschlTeller potential, our analysis points to the possibility of several rational extensions beyond those known in literature.
 Publication:

International Journal of Modern Physics A
 Pub Date:
 August 2015
 DOI:
 10.1142/S0217751X15501468
 arXiv:
 arXiv:1503.01394
 Bibcode:
 2015IJMPA..3050146S
 Keywords:

 Exactly solvable models;
 rational potentials;
 shape invariance;
 exceptional orthogonal polynomials;
 03.65.Ge;
 03.65.Sq;
 02.30.Hq;
 Solutions of wave equations: bound states;
 Semiclassical theories and applications;
 Ordinary differential equations;
 Mathematical Physics;
 Quantum Physics
 EPrint:
 18 pages, 1 figure