Conditionally exactly solvable potentials and exceptional orthogonal polynomials

Dutta, D.; Roy, Provas Kumar

doi:10.1063/1.3339676

Cited by 58 publications

(82 citation statements)

References 21 publications

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“…In 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.…”

Section: Introductionmentioning

confidence: 99%

Solvable rational extensions of the Morse and Kepler-Coulomb potentials

Grandati¹

2011

Journal of Mathematical Physics

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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...

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Section: Introductionmentioning

confidence: 99%

Solvable rational extensions of the Morse and Kepler-Coulomb potentials

Grandati¹

2011

Journal of Mathematical Physics

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“…In what follows, we shall present some examples of FPEs which are related to quantal systems whose ground states involve the recently discovered exceptional orthogonal polynomials [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. These can be viewed as deformed versions of some of the systems in [5].…”

Section: Equivalent Systemsmentioning

confidence: 99%

Extensions of a class of similarity solutions of Fokker-Planck equation with time-dependent coefficients and fixed/moving boundaries

Sasaki

2014

Journal of Mathematical Physics

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“…In view of the fact that the isospectral partner potentials have rather different type of eigenfunctions it is of interest to evaluate their position and momentum space entropies and compare them with the same of their partners. In this paper we shall consider the models of ref [18] and examine them from the point of view of EUR (1). …”

Section: Introductionmentioning

confidence: 99%

“…Interestingly exceptional orthogonal polynomials multiplied by square root of the associated weights appear as solutions of a number of exactly solvable potentials. Here we shall examine a class of conditionally exactly solvable potentials [15] which are isospectral to the harmonic oscillator potential [16,17] and whose solutions are given in terms of exceptional orthogonal polynomials related to the generalized Laguerre and Hermite polynomials [18].…”

Section: Introductionmentioning

confidence: 99%