Shape invariance and the SWKB series

Barclay, D.T.; Maxwell, C.J.

doi:10.1016/0375-9601(91)90869-a

Cited by 46 publications

(45 citation statements)

References 15 publications

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“…For all these, a 1 and a 0 are related by a translation. Careful analysis with this ansatz has failed to uncover any additional shape invariant potentials [10] and in fact it has been suggested that there are no others [11]. We shall now show that this is not the case since a large number of new shape invariant potentials can result from a new scaling ansatz…”

Section: Shape Invariance With Scaling Ansatzmentioning

confidence: 78%

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New exactly solvable Hamiltonians: Shape invariance and self-similarity

et al. 1993

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We discuss in some detail the self-similar potentials of Shabat and Spiridonov which are reflectionless and have an infinite number of bound states. We demonstrate that these self-similar potentials are in fact shape invariant potentials within the formalism of supersymmetric quantum mechanics. In particular, using a scaling ansatz for the change of parameters, we obtain a large class of new, reflectionless, shape invariant potentials of which the Shabat-Spiridonov ones are a special case.These new potentials can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for the energy eigenvalues, eigenfunctions and transmission coefficients for these potentials are obtained. We show that these potentials can also be obtained numerically. Included as an intriguing case is a shape invariant double well potential whose supersymmetric partner potential is only a single well. Our class of exactly solvable Hamiltonians is further enlarged by examining two new directions: (i) changes of parameters which are different from the previously studied cases of translation and scaling; (ii) extending the usual concept of shape invariance in one step to a multi-step situation. These extensions can be viewed as q-deformations of the harmonic oscillator or multi-soliton solutions corresponding to the Rosen-Morse potential.(

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Section: Shape Invariance With Scaling Ansatzmentioning

confidence: 78%

“…Careful analyses with this ansatz have failed to yield any additional shape invariant potentials [10]. Indeed it has been suggested [11] that there are no other shape invariant potentials. Although a rigorous proof has never been given, no counter examples have so far been found either.…”

Section: Introductionmentioning

confidence: 99%

New exactly solvable Hamiltonians: Shape invariance and self-similarity

et al. 1993

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“…Recent developments [7] have weakened this relation, but do not effect the conclusions in [4] concerning potentials for which the lowest-order SWKB condition is exact.…”

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confidence: 99%

“…We should point out immediately that there is a close connection here to the potentials discussed in [4] in connection with exact lowest-order SWKB quantisation conditions (see below). There equations (2) and (3) were satisfied by a superpotential φ, giving potentials…”

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confidence: 99%