Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

Abstract

Abstract. New exactly solvable rationally-extended 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)th-degree 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)th-degree Laguerre-type polynomials and a single one of (ν + 2)th-degree Jacobi-type 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.

The generalized quantum isotonic oscillator

“…Therefore, it cannot be used as a basis for an expansion. Similar sequences of polynomial eigenfunctions of a Sturm-Liouville problem have been found by Gómez-Ullate, Kamran and Milson [11] and by Quesne [12]. In the case of the first authors, the sequences does not include the constant (n = 0) polynomial.…”

Generalized Laguerre polynomials with position-dependent effective mass visualized via Wigner’s distribution functions

“…Schrödinger equation (SE) endowed with position-dependent effective mass (PDEM) has received a growing interest on behalf of physicists during the last decade [1,2,3,4,5,6,7,8,9,10,11,12]. Its solutions has been found to be very useful in describing, physically, the properties of the quantum dynamics of electrons in condensed matter physics as well as related fields of physics [13,14,15,16].…”

Disconjugacy, regularity of multi-indexed rationally extended potentials, and Laguerre exceptional polynomials

“…[11][12][13][14] The existence of two families of X m -Laguerre and X m -Jacobi EOP was also reported (thereby extending an observation made for m = 2 in Ref. 10) and it was later on explained through Darboux-Crum transformation. [15][16][17] In addition, these families were shown to be obtainable through two alternative approaches, namely, the Darboux-Bäcklund transformation (DBT) 18 and the prepotential method.…”

“…Since then, similar X m -Laguerre and X m -Jacobi EOP have been constructed and form the so-called third families. 10,18,19 A significant advance was then the introduction of multi-indexed X m 1 ,m 2 ,...,m k EOP families and associated rationally extended potentials through multistep Darboux algebraic transformations. 28 The same concept was developed through several approaches, such as the Crum-Adler mechanism, 29 higher order SUSYQM, 30,31 and multistep DBT.…”

“…In this context, the usefulness of an approach based on supersymmetric quantum mechanics (SUSYQM) or, equivalently, the Darboux transformation was pointed out. 9,10 An important step was the description of infinitely many shape-invariant potentials associated with X m EOP families, where the number of gaps m, called codimension (and corresponding here to n = 0, 1, . .…”