Solvable rational extensions of the Morse and Kepler-Coulomb potentials

Abstract

We show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential. This is made by using DarbouxBäcklund transformations based on unphysical regular Riccati-Schrödinger functions which are obtained from specific symmetries associated to the considered family of potentials. PACS numbers:I. II. INTRODUCTIONIn the recent years, several notable progresses have been made in the research and characterization of new closedform exactly solvable systems in quantum mechanics [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The obtained systems are regular rational extensions of some shape-invariant potentials [1][2][3] and are associated to families of exceptional orthogonal polynomials (EOP) built from the Laguerre or Jacobi classical orthogonal polynomials. In all the considered cases, the initial potentials belong to the second category (as defined in [22]) of primary translationally shape-invariant potentials (TSIP): the extended potentials of the J1 and J2 series (associated to the Jacobi EOP) are obtained from the generic second category potentials (Darboux-Pöschl-Teller or Scarf of the hyperbolic or trigonometric types), as for the extended potentials of the L1, L2 and L3 series, they are obtained from the unique exceptional second category potential which is the isotonic one.If we except the specific case of the harmonic oscillator which has been extensively treated [4,20,[23][24][25][26][27], the solvable extensions of first category potentials have been much less studied. Refering to the classification established in [22], the exceptional first category primary TSIP are the one-dimensional harmonic oscillator (HO), the Morse potential and the effective radial Kepler-Coulomb (ERKC) system, whereas the generic first category primary TSIP include the trigonometric and hyperbolic Rosen-Morse potentials as well as the Eckardt potential. A general study of the possible extensions of a large number of exactly solvable potentials from the point of view of conditionally solvable potentials has been made by Junker and Roy [35]. The case of the Morse potential has been also considered by Gomez-Ullate et al [4] who have determined the algebraic deformations of this system which are solvable by polynomials.In [21] we have developped a new approach which allows to generate an infinite set of regular exactly solvable extensions starting from every TSIP in a very direct and systematic way without taking recourse to any ansatz. This approach is based on a generalization of the usual SUSY partnership built from excited states. The corresponding Darboux-Bäcklund Transformations (DBT), which are covariance transformations for the class of Riccati-Schrödinger (RS) equations [22], are based on regularized RS functions which are obtained by using discrete symmetries acting on the parameters of the considered family of potentials. Considering the isotonic oscillator, we have obtained the three infinite sets L1, L2 an...

Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

“…[7], then reconsidered in Refs. [31] and [32], but that to the best of the author's knowledge, φ II A,m (x) has not been mentioned so far. To obtain some rationally-extended Morse potentials V A,B,ext (x) with given A and B, we have to start from a conventional Morse potential V A ′ ,B (x) with some different A ′ , but the same B (hence z remains unchanged).…”

“…1, the Coulomb potential has been rationally extended in two different ways by using the Darboux-Bäcklund transformation [31]. Our discovery of type II Morse extensions draws a parallel between the Morse and Coulomb potentials, which previously seemed compromised.…”

“…In the present work, we have reconsidered the construction of ES rationally-extended Morse potentials, previously carried out in the framework of the Darboux-Bäcklund transformation [31] or the prepotential method [32]. On using a first-order SUSYQM approach [1] and building on the concept of algebraic deformations of SI potentials [7], we have obtained the already known family of extended Morse potentials, corresponding to case iii of SUSYQM.…”

“…The latter expression was already given in previous studies [31,32], but the former is a new one, which has the advantage of directly providing us with the polynomial appearing in the first-excited state wavefunction,…”

“…In such a context, the Morse potential has been recently re-examined by making use of two different (but equivalent) approaches, the Darboux-Bäcklund transformation [31] and the prepotential method [32]. Both studies have recovered the previously found algebraic deformations [7].…”