Application of conditional shape invariance symmetry to obtain the eigen-spectrum of the mixed potentialV(r)=ar+br2+

Abstract

Application of conditional shape invariance symmetry to obtain the eigen-spectrum of the mixed potential V (r) = ar + br 2 + c r + l(l+1) AbstractWe show that the conditional shape invariance symmetry can be used as a very powerful tool to calculate the eigenvalues of the mixed potential V (r) = ar + br 2 + c r + l(l+1) r 2 for a restricted set of potential parameters. The energy for any state can be obtained algebraically, albeit for a severely restricted set of potential parameters. We also indicate that each member of the hierarchy of Hamiltonians is basically conditionally translational shape invariant. Comparison of analytically obtained results with numerical results is also presented. Our present methodology can be taken as an alternative treatment for the calculation of any higher order excited states of conditionally exactly solvable (CES) potentials.

An ubiquitous three-term recurrence relation

“…We test the accuracy of these results by means of the powerful Riccati-Padé method [17]. n,γ , W n,γ coming from the artificial truncation condition (15). These figures clearly confirm the theoretical analysis carried above.…”

“…In this paper we show that all those physical conclusions are mere artifacts of the unnecessary truncation condition that only produces some particular solutions without any physical relevance. The only interest in them is purely academic as they are exact solutions of a problem that is not exactly solvable [14,15]. Present numerical calculations, illustrated by means of three figures, reveal that there are square-integrable solutions for any value of the dimensionless parameter a (blue lines).…”

“…Present numerical results clearly show that there are square-integrable solutions for any B = B n,l . To be clearer: the set of bound states produced by the truncation condition (15) is contained in the set of all bound states. The majority of bound states do not satisfy such arbitrary, unnecessary condition.…”

“…and four for n = 3 It is obvious to anybody familiar with conditionally solvable quantum-mechanical models [14,15] (and references therein) that the approach just described does not produce all the eigenvalues of the operator L for a given set of values of γ and a but only those states with polynomial solutions for P (ξ). These particular eigenvalues W n,γ = 2n + 2|γ| + 2, n = 0, 1, .…”

“…The quantization of the parameter a (allowed values of a, or its dependence on the quantum numbers) is a mere artifact of the truncation condition (15) and does not exhibit any physical meaning. This fact is more than evident in the light of present numerical calculations.…”