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arXiv:quant-ph/9811088 · Physical Review A

Comment on “Conditionally exactly soluble class of quantum potentials”

Miloslav Znojil

Published May 17, 2000• 53 citations
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Abstract

We detect an omission in the paper ``Conditionally exactly soluble class of quantum potentials'' by A. de Souza Dutra [Phys. Rev. A 47, R2435 (1993)]. There, two strongly singular s-wave bound-state problems have been claimed completely solvable in closed form. Unfortunately, all the displayed wave functions are merely asymptotically correct solutions which do not satisfy the appropriate threshold boundary condition. We show that the incorporation of the threshold boundary conditions only leads to a very partial exact solvability at a single energy and for special couplings.

DOI
10.1103/physreva.61.066101

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The conditionally exactly solvable potentials: A misunderstanding

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In the paper "Conditionally exactly soluble class of quantum potentials" by A. de Souza Dutra [Phys. Rev. A 47 (1993) R2435] the whole $s-$wave spectrum of bound states in potentials $V_1(r)=A/r+B/r^{1/2}+G_0/r^2$ with $G_0=-3\\hbar/32\\mu$ and $V_2(r)=A r^{2/3}+B/r^{2/3}+g_0/r^2$ with $g_0=-5\\hbar^2/72\\mu$ has been constructed in closed form. We show that both the result and the method of its derivation are not correct.

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Reply to “Comment on ‘Conditionally exactly soluble class of potentials’ ”

ALVARO DE SOUZA DUTRA

We show that the problems raised by Znojil [Phys. Rev. A ▪, ▪ (2000)] preceding paper, can be resolved and that, in fact, there exist the proposed singular conditionally, exactly solvable potentials [Phys. Rev. A 47, R2435 (1993)].

Physical Review A•May 17, 2000•7 citations

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We detect an omission in the paper ``Conditionally exactly soluble class of quantum potentials'' by A. de Souza Dutra [Phys. Rev. A 47, R2435 (1993)]. There, two strongly singular s-wave bound-state problems have been claimed completely solvable in closed form. Unfortunately, all the displayed wave functions are merely asymptotically correct solutions which do not satisfy the appropriate threshold boundary condition. We show that the incorporation of the threshold boundary conditions only leads to a very partial exact solvability at a single energy and for special couplings.
In the paper "Conditionally exactly soluble class of quantum potentials" by A. de Souza Dutra [Phys. Rev. A 47 (1993) R2435] the whole s−s-s−wave spectrum of bound states in potentials V1(r)=A/r+B/r1/2+G0/r2V_1(r)=A/r+B/r^{1/2}+G_0/r^2V1​(r)=A/r+B/r1/2+G0​/r2 with G0=−3hbar/32muG_0=-3\\hbar/32\\muG0​=−3hbar/32mu and V2(r)=Ar2/3+B/r2/3+g0/r2V_2(r)=A r^{2/3}+B/r^{2/3}+g_0/r^2V2​(r)=Ar2/3+B/r2/3+g0​/r2 with g0=−5hbar2/72mug_0=-5\\hbar^2/72\\mug0​=−5hbar2/72mu has been constructed in closed form. We show that both the result and the method of its derivation are not correct.
We comment on several incorrect results obtained by Dong and Garcia-Ravelo (2007 Phys. Scr. 75 307) in the discussion of the bound-state solutions of the radial Schrodinger equation for the Manning–Rosen potential. We have identified the inconsistency in their reasoning on the allowed values of the parameter α and we have constructed the correct energy spectrum together with the normalized wavefunctions. Furthermore, we have shown that the wavefunction normalization constant can be obtained in closed form.
In this paper, the Born series for the s-wave scattering a0 is calculated for a class of central potentials V(r) up to sixth order in a dimensionless coupling strength g. Examples of exponentially decaying potentials as well as truncated potentials involving a single length-scale a are considered. In certain favorable cases, the exact result for the g-dependent s-wave scattering length a0=A0(g)a can be given in terms of special functions. The poles of A0(g) at increasing positive values of g correspond to the thresholds, where s-wave bound-states occur successively. Further scattering problems, where A0(g) is solvable in terms of elementary functions, are also presented.