Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
TLDR
In this paper, the authors construct two new exactly solvable potentials giving rise to bound-state solutions to the Schrodinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type X1 exceptional orthogonal polynomials.Abstract:
We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schrodinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type X1 exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.read more
Citations
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An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
TL;DR: In this article, two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem are presented, and they are shown to be orthogonal with respect to a positive definite inner product defined over the compact interval [ − 1, 1 ] or the half-line [ 0, ∞ ), respectively.
Journal ArticleDOI
Infinitely many shape invariant potentials and new orthogonal polynomials
Satoru Odake,Ryu Sasaki +1 more
TL;DR: In this paper, the shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic Poschl-Teller potentials in terms of their degree l polynomial eigen functions are presented.
Journal ArticleDOI
Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials
Satoru Odake,Ryu Sasaki +1 more
TL;DR: In this paper, infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems, where each polynomial has another integer n which counts the nodes and the totality of the integer indices correspond to the degrees of the virtual state wavefunctions which are deleted by the generalisation of Crum-Adler theorem.
Journal ArticleDOI
Another set of infinitely many exceptional (X_{\ell}) Laguerre polynomials
Satoru Odake,Ryu Sasaki +1 more
TL;DR: Odake et al. as mentioned in this paper presented a new set of infinitely many shape invariant potentials and the corresponding exceptional (X l ) Laguerre polynomials.
Journal ArticleDOI
Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux–Crum transformations
TL;DR: In this article, a simple derivation is presented of the four families of infinitely many shape-invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials.
References
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Dynamical Breaking of Supersymmetry
TL;DR: In this article, general conditions for dynamical supersymmetry breaking are discussed and examples are given (in 0 + 1 and 2 + 1 dimensions) in which such a program in four dimensions is possible.
Journal ArticleDOI
Supersymmetry and quantum mechanics
TL;DR: In this article, the authors review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications, including shape invariance and operator transformations, and show that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials.
Journal ArticleDOI
The factorization method
Leopold Infeld,T. E. Hull +1 more
TL;DR: The first-order differential-difference factorization method as mentioned in this paper is an operational procedure which enables us to answer, in a direct manner, questions about eigenvalue problems which are of importance to physicists.
Journal Article
Derivation of Exact Spectra of the Schrodinger Equation by Means of Supersymmetry
Journal ArticleDOI
An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
TL;DR: In this article, two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem are presented, and they are shown to be orthogonal with respect to a positive definite inner product defined over the compact interval [ − 1, 1 ] or the half-line [ 0, ∞ ), respectively.
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