scholarly journals Quasi‐exactly solvable systems and orthogonal polynomials

1996 ◽  
Vol 37 (1) ◽  
pp. 6-11 ◽  
Author(s):  
Carl M. Bender ◽  
Gerald V. Dunne
Keyword(s):  
Orthogonal Polynomials ◽  
Exactly Solvable

scholarly journals Do quasi-exactly solvable systems always correspond to orthogonal polynomials?

1998 ◽  
Vol 239 (4-5) ◽  
pp. 197-200 ◽  
Author(s):  
Avinash Khare ◽  
Bhabani Prasad Mandal
Keyword(s):  
Orthogonal Polynomials ◽  
Exactly Solvable

scholarly journals LIE ALGEBRAIC DISCRETIZATION OF DIFFERENTIAL EQUATIONS

1995 ◽  
Vol 10 (24) ◽  
pp. 1795-1802 ◽  
Author(s):  
YURI SMIRNOV ◽  
ALEXANDER TURBINER
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Heisenberg Algebra ◽  
Difference Operators ◽  
Discrete Version ◽  
Classical Orthogonal Polynomials ◽  
Finite Difference Equations ◽  
Exactly Solvable

A certain representation for the Heisenberg algebra in finite difference operators is established. The Lie algebraic procedure of discretization of differential equations with isospectral property is proposed. Using sl 2-algebra based approach, (quasi)-exactly-solvable finite difference equations are described. It is shown that the operators having the Hahn, Charlier and Meissner polynomials as the eigenfunctions are reproduced in the present approach as some particular cases. A discrete version of the classical orthogonal polynomials (like Hermite, Laguerre, Legendre and Jacobi ones) is introduced.

scholarly journals HIGHER-ORDER SUSY, EXACTLY SOLVABLE POTENTIALS, AND EXCEPTIONAL ORTHOGONAL POLYNOMIALS

2011 ◽  
Vol 26 (25) ◽  
pp. 1843-1852 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Higher Order ◽  
Wave Functions ◽  
Supersymmetric Quantum Mechanics ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Exceptional Orthogonal Polynomials

Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial gμ arising in the denominator of the potentials.

scholarly journals RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM

2011 ◽  
Vol 26 (32) ◽  
pp. 5337-5347 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Laguerre Polynomials ◽  
Supersymmetric Quantum Mechanics ◽  
Shape Invariance ◽  
Order Analysis ◽  
First Order ◽  
Exactly Solvable ◽  
Exceptional Orthogonal Polynomials ◽  
Differential Relations

A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to kth-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a (k-1)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invariance.

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