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The conditionally exactly solvable potentials: A misunderstanding

Miloslav Znojil

Published Nov 30, 1998• 0 citations
OpenAlex

Abstract

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.

Related papers

Comment on “Conditionally exactly soluble class of quantum potentials”

Miloslav Znojil

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.

arXiv:quant-ph/9811088•Physical Review A•May 17, 2000•53 citations

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

Conditionally exactly solvable potential and dual transformation in quantum mechanics

Bijan Kumar Bagchi, C. Quesne

We show that the conditionally exactly solvable potential of Dutt et al (1995 J. Phys. A: Math. Gen. 28 L107) and the exactly solvable potential from which it is derived form a dual system.

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Comment on “Coherent states for exactly solvable potentials”

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In Phys. Rev. A 69, 012102 (2004), Schreecharan, Panigrahi, and Banerji present a general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials, e.g., Morse and Poschl-Teller. I have examined coherent states of the Morse oscillator presented by the authors and have found them to have incorrect mathematical substitutions. As such, this Comment looks at the correctness of two important equations that have led to their Morse oscillator coherent states.

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Expanding the class of conditionally exactly solvable potentials

ALVARO DE SOUZA DUTRA, Uwe Girlich

Recently a class of quantum-mechanical potentials was presented that is characterized by the fact that they are exactly solvable only when some of their parameters are fixed to a convenient value, so they were christened as conditionally exactly solvable potentials. Here we intend to expand this class by introducing examples in two dimensions. As a byproduct of our search, we found also another exactly solvable potential.

Physical Review A•Nov 1, 1994•4 citations

Study on conditions for the existence of bound states in central field with the variational method

Chen Gang

In quantum physics,the variational method can be used to study the conditions for the existence of bound states in a central field.We take the following six kinds of potentials,V1(r)=-V0a δ(r-a),V2(r)=-V0e-r/a,V3(r)=-V0ae-r/a/r,V4(r)=-V0e-r2/a2,V5(r)=-e2/r,and V6(r)=-A/r2 as examples to demonstrate the usage of this method.The derived conditions for V1 to V4 are μγa2/h-20.68,0.84,1,1.95 respectively;for V5,the bound states can exist for e with arbitrary value,while for V6,there is no bound states for any value of A.We also get the precise conditions by solving the Schroedinger equation for these potentials.A comparison of the results obtained by these two methods shows that the conditions derived by the variational method are sufficient,but not necessary conditions.

Journal of Shandong University•Jan 1, 2009•0 citations
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 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 quantum physics,the variational method can be used to study the conditions for the existence of bound states in a central field.We take the following six kinds of potentials,V1(r)=-V0a δ(r-a),V2(r)=-V0e-r/a,V3(r)=-V0ae-r/a/r,V4(r)=-V0e-r2/a2,V5(r)=-e2/r,and V6(r)=-A/r2 as examples to demonstrate the usage of this method.The derived conditions for V1 to V4 are μγa2/h-20.68,0.84,1,1.95 respectively;for V5,the bound states can exist for e with arbitrary value,while for V6,there is no bound states for any value of A.We also get the precise conditions by solving the Schroedinger equation for these potentials.A comparison of the results obtained by these two methods shows that the conditions derived by the variational method are sufficient,but not necessary conditions.