Revisiting (quasi-)exactly solvable rational extensions of the morse potential
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TL;DR: In this paper, the construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics, and the existence of another family of extended potentials, strictly isospectral to VA+1, B(x), is pointed out.
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Abstract: The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials VA, B, ext(x), obtained from a conventional Morse potential VA-1, B(x) by the addition of a bound state below the spectrum of the latter, is reobtained. More importantly, the existence of another family of extended potentials, strictly isospectral to VA+1, B(x), is pointed out for a well-chosen range of parameter values. Although not shape invariant, such extended potentials exhibit a kind of "enlarged" shape invariance property, in the sense that their partner, obtained by translating both the parameter A and the degree m of the polynomial arising in the denominator, belongs to the same family of extended potentials. The point canonical transformation connecting the radial oscillator to the Morse potential is also applied to exactly solvable rationally-extended radial oscillator potentials to build quasi-exactly solvable rationally-extended Morse ones.
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Citations
Krein-Adler transformations for shape-invariant potentials and pseudo virtual states
Satoru Odake,Ryu Sasaki +1 more
TL;DR: In this paper, the Darboux-Crum transformation in terms of multiple pseudo virtual state wavefunctions is shown to be equivalent to Krein-Adler transformations deleting multiple eigenstates with shifted parameters.
83
Krein–Adler transformations for shape-invariant potentials and pseudo virtual states
Satoru Odake,Ryu Sasaki +1 more
TL;DR: In this article, the Darboux-Crum transformation in terms of multiple pseudo virtual state wavefunctions is shown to be equivalent to the Krein-Adler transformation, deleting multiple eigenstates with shifted parameters.
70
Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators
Ian Marquette,Christiane Quesne +1 more
TL;DR: In this paper, the type III Hermite Xm exceptional orthogonal polynomial family is generalized to a double-indexed one and corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics.
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Extensions of solvable potentials with finitely many discrete eigenstates
Satoru Odake,Ryu Sasaki +1 more
TL;DR: In this paper, rational extensions of six examples of shape-invariant potentials having finitely many discrete eigenstates are studied. But their degrees are much higher than nmax so that their energies are lower than the groundstate energy.
59
Recurrence relations of the multi-indexed orthogonal polynomials
TL;DR: In this article, the bispectral properties of Laguerre and Jacobi polynomials of the Hermite type are discussed and a method to obtain the coefficients of the recurrence relations explicitly is presented.
References
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.
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Quasi-Exactly Solvable Models in Quantum Mechanics
Alex Ushveridze
- 01 Jan 1994
TL;DR: In this paper, the Lanczos tridiagonalization procedure was used to construct quasi-exactly solvable models with separable variables, and the Gelfand-Levitan equation was used for the first time.
755
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.
458
Factorization method and new potentials with the oscillator spectrum
TL;DR: In this article, a one-parameter family of potentials in one dimension was constructed with the energy spectrum coinciding with that of the harmonic oscillator, which is a new derivation of a class of possible potentials previously obtained by Abraham and Moses with the help of the Gelfand-Levitan formalism.
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