Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
Abstract
New exactly solvable rationallyextended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g. The cases where g is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same characteristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are constructed. In the linear case, they contain (ν+1)thdegree polynomials with ν = 0,1,2,..., which are shown to be X_{1}Laguerre or X_{1}Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of (ν+2)thdegree Laguerretype polynomials and a single one of (ν+2)thdegree Jacobitype polynomials with ν = 0,1,2,... are identified. They are candidates for the still unknown X_{2}Laguerre and X_{2}Jacobi exceptional orthogonal polynomials, respectively.
 Publication:

SIGMA
 Pub Date:
 August 2009
 DOI:
 10.3842/SIGMA.2009.084
 arXiv:
 arXiv:0906.2331
 Bibcode:
 2009SIGMA...5..084Q
 Keywords:

 Schrödinger equation;
 exactly solvable potentials;
 supersymmetry;
 orthogonal polynomials;
 Mathematical Physics;
 Quantum Physics
 EPrint:
 v2: additions in secs. 1 and 4, 5 new references