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.

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 Exactly and quasi-exactly solvable ‘discrete’ quantum mechanics

2011 ◽  
Vol 369 (1939) ◽  
pp. 1301-1318 ◽  
Author(s):  
Ryu Sasaki
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Essential Role ◽  
Dynamical Symmetry ◽  
Oscillator Algebra ◽  
Discrete Variable ◽  
Shape Invariance ◽  
One Dimensional ◽  
Exactly Solvable ◽  
Creation Operators

A brief introduction to discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators and dynamical symmetry algebras, including the q -oscillator algebra and the Askey–Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable (QES) Hamiltonians in one-dimensional ‘discrete’ quantum mechanics is presented. It reproduces all the known Hamiltonians whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and QES Hamiltonians are constructed. The sinusoidal coordinate plays an essential role.

CONDITIONALLY EXACTLY SOLVABLE SINGULAR EVEN POWER POTENTIAL IN SUPERSYMMETRIC QUANTUM MECHANICS

2002 ◽  
Vol 17 (21) ◽  
pp. 1367-1375 ◽  
Author(s):  
BARNALI CHAKRABARTI ◽  
TAPAN KUMAR DAS
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Riccati Equation ◽  
Schrodinger Equation ◽  
Supersymmetric Quantum Mechanics ◽  
New Technique ◽  
Body System ◽  
Operator Formalism ◽  
Shape Invariance ◽  
Exactly Solvable

We solve strongly singular even power potential with inverse quadratic, inverse quartic and inverse sextic terms, using superpotential ansatz technique. We show that our new technique is very powerful and simple compared to traditional wave function ansatz technique. We also point out conditional shape invariance as the underlying symmetry to get conditional exactness. We also generalize the case for two-body Hamiltonian and propose an alternative way to solve Riccati equation instead of solving Schrödinger equation. We present operator formalism which is rather powerful. We also discuss the problem encountered in generalizing the potential for an N-body system.

scholarly journals SUSUSY Quantum Mechanics

1997 ◽  
Vol 12 (01) ◽  
pp. 171-176 ◽  
Author(s):  
David J. Fernández C.
Keyword(s):  
Quantum Mechanics ◽  
Shift Operator ◽  
Second Order ◽  
Parametric Family ◽  
First Order ◽  
Shift Operators ◽  
Exactly Solvable ◽  
Exactly Solvable Hamiltonians

The exactly solvable eigenproblems in Schrödinger quantum mechanics typically involve the differential "shift operators". In the standard supersymmetric (SUSY) case, the shift operator turns out to be of first order. In this work, I discuss a technique to generate exactly solvable eigenproblems by using second order shift operators. The links between this method and SUSY are analysed. As an example, we show the existence of a two-parametric family of exactly solvable Hamiltonians, which contains the Abraham–Moses potentials as a particular case.

scholarly journals Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1853
Author(s):  
Christiane Quesne
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Supersymmetric Quantum Mechanics ◽  
Invariance Condition ◽  
Partial Differential ◽  
Shape Invariance ◽  
Shape Invariant ◽  
Infinite Set

We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial differential equations, can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ.

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