Infinitely many shape invariant potentials and new orthogonal polynomials

Одаке, Сатору; Sasaki, Ryu

doi:10.1016/j.physletb.2009.08.004

Cited by 222 publications

(370 citation statements)

References 17 publications

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“…After the introduction of the first families of exceptional orthogonal polynomials (EOP) in the context of Sturm-Liouville theory [11,12], the realization of their usefulness in constructing new SI extensions of ES potentials in quantum mechanics [13,14,15], and the rapid developments that followed in this area [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], it soon appeared that only some of the well-known SI potentials led to rational extensions connected with EOP. In this category, one finds the radial oscillator [13,15,16,17,18,22,23,24], the Scarf I (also called trigonometric Pöschl-Teller or Pöschl-Teller I) [13,15,16,17,22,24], and the generalized Pöschl-Teller (also termed hyperbolic Pöschl-Teller or Pöschl-Teller II) [14,16,17].…”

Section: Introductionmentioning

confidence: 99%

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Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

Quesne

2012

Int. J. Mod. Phys. A

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The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials V A,B,ext (x), obtained from a conventional Morse potential V A−1,B (x) by the addition of a bound state below the spectrum of the latter, is re-obtained. More importantly, the existence of another family of extended potentials, strictly isospectral to V A+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. Running head: Rational Extensions of Morse Potential

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Section: Introductionmentioning

confidence: 99%

“…In this category, one finds the radial oscillator [13,15,16,17,18,22,23,24], the Scarf I (also called trigonometric Pöschl-Teller or Pöschl-Teller I) [13,15,16,17,22,24], and the generalized Pöschl-Teller (also termed hyperbolic Pöschl-Teller or Pöschl-Teller II) [14,16,17].…”

Section: Introductionmentioning

confidence: 99%

Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

Quesne

2012

Int. J. Mod. Phys. A

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“…, m − 1), may be arbitrarily large. [11][12][13][14] The existence of two families of X m -Laguerre and X m -Jacobi EOP was also reported (thereby extending an observation made for m = 2 in Ref. 10) and it was later on explained through Darboux-Crum transformation.…”

Section: Introductionmentioning

confidence: 65%

Disconjugacy, regularity of multi-indexed rationally extended potentials, and Laguerre exceptional polynomials

Grandati

Quesne

2013

Journal of Mathematical Physics

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The power of the disconjugacy properties of second-order differential equations of Schrödinger type to check the regularity of rationally extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re-examining the extensions of the isotonic oscillator (or radial oscillator) potential derived in kth-order supersymmetric quantum mechanics or multistep Darboux-Bäcklund transformation method. The function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term, whose value is calculated by induction over k. The sign of this term being the same as that of the already known highest degree term, the potential denominator has the same sign at both extremities of the definition interval, a property that is shared by the seed eigenfunction used in the potential construction. By virtue of disconjugacy, such a property implies the nodeless character of both the eigenfunction and the resulting potential. C 2013 AIP Publishing LLC.

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“…where the parameters V 1 and V 2 describe the property of the potential well while the parameter α is related to the range of this potential [23,25,26]. In Figure 1, we draw the trigonometric PT potential (1) for parameter values V 1 = 5.0 fm −1 , V 2 = 3.0 fm −1 , and α = 0.8 fm −1 .…”

Section: Introductionmentioning

confidence: 99%

Approximate Solution of the Spin-0 Particle Subject to the Trigonometric Pöschl–Teller Potential with Centrifugal Barrier

Hamzavi

Rajabi

Amirfakhrian

2013

Zeitschrift Für Naturforschung A

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The trigonometric Pöschl-Teller (PT) potential describes the diatomic molecular vibration. In this paper, we study the approximate solutions of the radial Klein-Gordon (KG) equation for the rotating trigonometric PT potential using the Nikiforov-Uvarov (NU) method. The energy eigenvalues and their corresponding eigenfunctions are calculated for arbitrary l-states in closed form. We obtain the non-relativistic limit and present some numerical results for both relativistic and non-relativistic cases.

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Infinitely many shape invariant potentials and new orthogonal polynomials

Cited by 222 publications

References 17 publications

Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

Disconjugacy, regularity of multi-indexed rationally extended potentials, and Laguerre exceptional polynomials

Approximate Solution of the Spin-0 Particle Subject to the Trigonometric Pöschl–Teller Potential with Centrifugal Barrier

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