Supersymmetric solutions of PT-/non-PT-symmetric and non-Hermitian Screened Coulomb potential via Hamiltonian hierarchy inspired variational method

Gholamreza Faridfathi; Ramazan Sever

REVIEW

Supersymmetric solutions of PT-/non-PT-symmetric and non-Hermitian Screened Coulomb potential via Hamiltonian hierarchy inspired variational method

Gholamreza Faridfathi, Ramazan Sever

Quantum Physics · Submitted 23 Feb 2006 · 40 references

Pith's verdict: The advertised variational derivation does not actually produce Eq. (21), and Table 1 contradicts the accuracy claim, so the paper is not publishable in its current form.

T0 · AI review

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arXiv IDquant-ph/0602194 (version 1)

Pith IDQCZL4BCX, Pith's permanent record of this paper and its review

CitationGholamreza Faridfathi, Ramazan Sever. "Supersymmetric solutions of PT-/non-PT-symmetric and non-Hermitian Screened Coulomb potential via Hamiltonian hierarchy inspired variational method." arXiv:quant-ph/0602194 (2006). https://pith.science/paper/QCZL4BCX

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Claim paper

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Short version: this paper advertises a Hamiltonian-hierarchy variational derivation of ECSC bound-state energies, but the central formula is not derived from the integrals on display, and Table 1 contradicts the accuracy claim. It is not publishable as it stands.

Where credit is due: the manuscript is honest about its limitations. It states explicitly that the proposed superpotentials satisfy the Riccati equation only for effective Hulthen-type potentials, not for the actual ECSC potential. For mu=0, Eq. (21) reduces correctly to the known Hulthen/Yukawa formula. The review sections are readable.

The soft spots are serious. Eqs. (12) and (16) display Rayleigh quotients, but those integrals are never evaluated, the minimization over mu1 and mu2 is never performed, and the minimizers are never reported. Worse, alpha and beta are complex, so the expectation values are complex and a variational minimum is not even defined unless the real and imaginary parts are separated, which the paper does not do. Eq. (21) is instead read off the constant terms of the effective potentials (19)-(20), which is a different operation. The paper itself admits the superpotentials are not solutions of the original Riccati equation. And the only numerical evidence, Table 1, shows 'Our Work' energies for 2p, 3d, and 4f roughly a factor of two below the exact values (e.g., 2p at lambda=0.02: -0.105890 vs -0.211900). That directly contradicts the 'approximate yet accurate' claim. The PT-symmetric and non-Hermitian sections simply assert the same energies without new computation.

Novelty is modest. The ansatz is a variation of Hulthen-type superpotentials already used for screened Coulomb potentials; the paper cites its own earlier work [36] on that, which is fine, but this paper adds the cosine factor by substitution rather than by solving a new problem. The citation pattern looks clean.

Who is this for? Maybe someone collecting Hulthen-type approximations. But the central claim needs repair. I would not send this to peer review unless the authors actually evaluate the variational integrals, report minimizers, and reconcile Table 1, or reframe it as an effective Hulthen approximation with error estimates. As it stands, desk reject is the right call.

Referee report

Major objections (5)

[§3, Eqs. (12) and (16)] The central claim that Eq. (21) is obtained by a variational minimization is not supported by the manuscript. The trial wave functions Ψ_{μ1}(r) and Ψ_{μ2}(r) are written down, and the variational energies E_{μ1} and E_{μ2} are defined as Rayleigh quotients, but the integrals are never evaluated and the minimization with respect to μ1 and μ2 is never performed. The optimizing values μ̅1 and μ̅2 are not reported, nor is the value of the minimized energy. Without these steps, Eq. (21) cannot be said to follow from the variational principle stated.
[§3, Eqs. (17)-(20)] The paper explicitly acknowledges that the proposed superpotentials do not satisfy the Riccati equation for the actual ECSC potential, but only for 'effective potentials' V1eff and V2eff. These effective potentials are Hulthen-type potentials whose constant terms encode the known Hulthen spectrum. Equation (21) is obtained by substituting α and β into those constant terms, not by evaluating the expectation values (12) and (16). No argument is given that the energies of V1eff and V2eff, even after optimization, approximate the energies of the original ECSC potential; the text simply asserts this. Since the entire derivation rests on this unverified replacement, the derivation of Eq. (21) is incomplete.
[Table 1] The numerical evidence presented in Table 1 does not support the claimed 'approximate yet accurate results.' For the 2p state at λ=0.02, the 'Our Work' value is -0.105890 while the exact numerical value is -0.211900, a factor-of-two discrepancy that is far beyond any reasonable approximation error for a bound-state energy. The same pattern holds for 3d (λ=0.02: -0.0375 vs -0.07503) and 4f (λ=0.02: -0.0147 vs -0.02897). These systematic errors contradict the claim of accuracy and indicate that Eq. (21) does not provide a valid estimate for these states.
[§3, Eqs. (12) and (16) with complex α and β] For the real ECSC potential, α = λ - iμ and β = λ + iμ are complex. The trial wave functions in Eqs. (11) and (15) are therefore complex, and the expectation values in Eqs. (12) and (16) are generally complex numbers. A variational minimum is not well defined for a complex expectation value unless the real and imaginary parts are separated and minimized independently; the manuscript does not do this. This undermines the claim that 'minimizing the energies E_{μ1} and E_{μ2}' yields real bound-state energies in the real ECSC case.
[§§4-5, Eqs. (23)-(31)] The extensions to the non-PT-symmetric and PT-symmetric non-Hermitian cases are asserted rather than derived. The authors state that 'following the same method will yield the same energy eigenvalues as in (21),' but no variational integrals, minimization conditions, or reality conditions on parameters are discussed. Since the underlying variational procedure is already not demonstrated for the real potential, the claim that the identical spectrum holds for these complex variants is unsupported.

Minor objections (4)

[Throughout] There are numerous typographical and formatting errors, including 'Schr¨ odinger,' 'eig enfunctions,' and 'the the Hamiltonian hierarchy method,' which should be corrected.
[Table 1] Several entries in Table 1 appear to be typographical errors: the 1s λ=0.100 'NR-QM Variational' entry is listed as -0.470600 while the exact value is around -0.407000, and the 4f λ=0.080 'Our Work' entry is positive (-0.184500), which is unphysical for a bound-state energy. These should be corrected or flagged.
[§3, Eq. (22)] For μ=0, Eq. (21) reduces to Eq. (22). The limiting case should be checked against the known Yukawa potential spectra; the manuscript does not discuss whether the limit is consistent with established results beyond a formal substitution.
[References] Several references are cited in the context of the numerical comparison (e.g., [20], [39], [40]) but the Table does not clearly identify which method generated each column. The 'SUSYQM Our Work' label is unclear, and the paper does not explain how the 'Exact Numerical' values from [39] were computed.

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The paper

The paper claims

A Hulthen-type superpotential ansatz yields a single closed formula for the bound-state energies of the exponential-cosine screened Coulomb potential and its PT-symmetric and non-Hermitian variants.

The reading

The paper sets out to obtain approximate bound-state energies and wave functions for the exponential-cosine screened Coulomb (ECSC) potential, $V(r)=-(q/r)e^{-\lambda r}\cos(\mu r)$, by proposing Hulthen-style superpotentials inside a Hamiltonian-hierarchy variational scheme. It claims that the energies reduce to the closed formula $E=-(q/2)[1/(l+1)^2+(\lambda^2-\mu^2)/4-\lambda/(l+1)]$, with Table 1 offered as evidence that the results are approximate yet accurate. The same formula is asserted to hold for the non-PT-symmetric non-Hermitian version and the PT-symmetric non-Hermitian version of the potential. If the claim is right, it would supply a simple analytic estimate for screened Coulomb systems in atomic, nuclear, and plasma settings where the Schrödinger equation for this potential is not exactly solvable.

What carries the argument

The load-bearing object is the Hulthen-type superpotential $W_{(l+1)}(r)=-(l+1)\alpha e^{-\alpha r}/(1-e^{-\alpha r})+1/(l+1)-\alpha/2$, one for each of the two complex exponentials in the ECSC potential. It is inserted into the Riccati equation not for the actual ECSC potential but for the effective potentials (19) and (20); the constant terms of those effective potentials are then identified with the bound-state energy, and trial eigenfunctions are exponentials of the integral of the superpotential. The replacement of $\alpha$ and $\beta$ by variational parameters $\mu_1$ and $\mu_2$ is the variational part of the scheme, although the integrals (12) and (16) and the minimizing parameter values are not shown in the paper.

Extended reading notes

Core claim

The central claim is that the bound-state energies of the ECSC potential can be read off from the constant terms of two effective potentials built from a Hulthen-like superpotential ansatz. For the two complex Yukawa pieces $V_1(r)=-e^{-(\lambda-i\mu)r}/r$ and $V_2(r)=-e^{-(\lambda+i\mu)r}/r$, the paper proposes superpotentials of the form $W_{(l+1)}(r)=-(l+1)\alpha e^{-\alpha r}/(1-e^{-\alpha r})+1/(l+1)-\alpha/2$ and the analogous partner, substitutes each into the Riccati equation, and obtains effective potentials whose additive constants give $E=-(q/2)[1/(l+1)^2+(\lambda^2-\mu^2)/4-\lambda/(l+1)]$. It asserts that the same energy formula applies when the potential is complexified into non-PT and PT-symmetric non-Hermitian forms, and that the corresponding superpotentials and ground-state wave functions satisfy the stated symmetry conditions.

Load-bearing premise

The method assumes that the energies of the simplified stand-in potentials built from the superpotential equal the energies of the actual exponential-cosine screened Coulomb potential, even though the tuning parameters that would make that match are never reported.

Editorial extensions

If this is right

For $\mu=0$, the formula reduces to $E=-(q/2)[1/(l+1)-\lambda/2]^2$, the Yukawa screened Coulomb limit, giving an immediate check against existing results.
The same energy expression is claimed for the Hermitian, non-PT-symmetric non-Hermitian, and PT-symmetric non-Hermitian versions, so PT-symmetric complexification would not shift the bound-state levels.
The energy depends on the cosine modulation only through $\lambda^2-\mu^2$, so the oscillatory part enters the energy quadratically and with the opposite sign from $\lambda^2$.
If accurate, the closed formula gives quick estimates for low-lying states such as $1s$, $2p$, $3d$, and $4f$ across screening parameters without numerical integration.
The same Riccati-constant strategy could be reused for other non-exactly-solvable exponential-type potentials, as the paper suggests.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the paper never evaluates the variational integrals (12) and (16) or reports the minimizing parameters, a natural next step is to carry out those integrals: if their minimum reproduces Eq. (21), the variational step is closed, and a numerical Schrödinger solver can then test the effective-potential identification directly.
The formula's dependence on $\mu$ only through $\mu^2$ implies an unstated symmetry: states with $\mu$ and $-\mu$ would have the same energy, a prediction a direct numerical diagonalization could confirm or refute.
If the identification were confirmed, the same Hulthen-ansatz route could be transferred to other complexified screened potentials, such as exponential-sine or hyperbolic-cosine variants, yielding closed-form estimates where no exact solution is known.
A separate numerical check would be needed to see whether the claimed accuracy extends beyond the $1s$ row of Table 1, since the comparison columns for the other angular-momentum rows are incomplete or show larger gaps.

Original abstract

The supersymmetric solutions of PT-symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schrodinger equation for the Exponential-Cosine Screened Coulomb potential. The Hamiltonian hierarchy inspired variational method is used to obtain the approximate energy eigenvalues and corresponding wave functions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,332 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.19 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The supersymmetric solutions of PT-symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schrodinger equation for the Exponential-Cosine Screened Coulomb potential. The Hamiltonian hierarchy inspired variational method is used to obtain the approximate energy eigenvalues and corresponding wave functions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The supersymmetric solutions of PT-symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schrodinger equation for the Exponential-Cosine Screened Coulomb potential.	contribution 0.74	0.86	0.36	asserted 1.00
s1	The Hamiltonian hierarchy inspired variational method is used to obtain the approximate energy eigenvalues and corresponding wave functions.	method 1.00	0.89	0.13	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.91 · evidence from proof or derivation 0.80.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (40)

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[40]

R. Sever, C. Tezcan, Tr. J. of Physics, 17 (1993) 459-464. 12 Table 1 : Energy eigenvalues of ECSC as a function of the screening p arameter λ for 1 s, 2p, 3 d and 4 f states in Rydberg units of energy. State 1 s Screening λ SUSYQM Our Work Hypervirial Solution [40 ] NR − QM Variational [20 ] Exact Numerical [39 ] 0.020 -0.480290 -0.480310 -0.480300 -0.48...

View on Pith 1993

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [36]

G. Faridfathi, R. Sever, Metin Akta¸ s, Journal of Mathematical Chemistry Vol. 38, No.4, (2005) 533-540

View on Pith 2005

[2] [1]

F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251 (1995) 267

View on Pith 1995

[3] [2]

Richard W. Haymaker, A. R. P. Rau, Am. J. Phys. 54 (1986) 928

View on Pith 1986

[4] [3]

E. Drigo Filho, R. Maria Ricota, Phys. Lett. A, 299 (2002) 137

View on Pith 2002

[5] [4]

L. Gedenshtein, I. V. Krive, Soviet Phys. Usp. 28 (1985) 645

View on Pith 1985

[6] [5]

G. L´ evai, in: H.V. von Gevamb (Ed.), Lecture Notes in Phys., Vol. 427, Springer (1993) 427

View on Pith 1993

[7] [6]

E. Drigo Filho, Mod. Phys. Rev. A, 9 (1994) 411

View on Pith 1994

[8] [7]

E. Drigo. Filho, R. Maria Ricota, Phys. Atom. Nucl. 61 (1998) 1836

View on Pith 1998

[9] [8]

E. Drigo Filho, R. Maria Ricota, Mod. Phys. Lett. A, 10 (1995) 1613

View on Pith 1995

[10] [9]

E. Drigo Filho, R. Maria Ricota, Mod. Phys. Lett. A, 15 (2000) 1253

View on Pith 2000

[11] [10]

I. B. Golberg, R. H. Pratt, J. Math. Phys. 28 (1987) 1351

View on Pith 1987

[12] [11]

O. V. Gabriel, S. Chaudhuri, R. H. Pratt, Phys. Rev. A, 24 (1981) 3088

View on Pith 1981

[13] [12]

S. J. R. Crossley, Adv. At. Mol. Phys. 5 (1969) 237

View on Pith 1969

[14] [13]

E. R. Vrscay, H. Hamidian, Phys. Lett. A, 130 (1988) 141

View on Pith 1988

[15] [14]

C. K. Au, Y. Aharonov, Phys. Rev. A, 20 (1979) 2245

View on Pith 1979

[16] [15]

C. K. Au, G. W. Rogers, Phys. Rev. A, 22 (1980) 1820

View on Pith 1980

[17] [16]

G. W. Rogers, Phys. Rev. A, 30 (1984) 35

View on Pith 1984

[18] [17]

A. V. Sargeev, A. I. Sherstyuk, Sov. Nucl. Phys., 39 (1984) 731. 10

View on Pith 1984

[19] [18]

C. H. Mehta, S. H. Patil, Phys. Rev. A, 17 (1978) 34

View on Pith 1978

[20] [19]

M. L. Du, Phys. Lett. A, 133 (1988) 109

View on Pith 1988

[21] [20]

C. S. Lam, Y. P. Varshni, Phys. Rev. A, 4 (1971) 1875

View on Pith 1971

[22] [21]

M. Znojil, Phys. Lett. A, 102 (1984) 289

View on Pith 1984

[23] [22]

F. D. Adame, A. Rodriguez, Phys. Lett. A, 198 (1995) 275

View on Pith 1995

[24] [23]

N. A. Rao, B. A. Kagali, Phys. Lett. A, 296 (2002) 192

View on Pith 2002

[25] [24]

M. M. Panja, R. Dutt, Y. P. Varshni, Phys. Rev. A, 42 (1990) 106

View on Pith 1990

[26] [25]

A. Chatterjee, Phys. Rev. A, 35 (1987) 2722

View on Pith 1987

[27] [26]

R. Sever, C. Tezcan, Phys. Rev. A, 35 (1987) 2725; ibid, 36 (1987) 1045

View on Pith 1987

[28] [27]

G. M. Harris, Phys. Rev., 125 (1962) 1131

View on Pith 1962

[29] [28]

C. R. Smith, Phys. Rev. A, 134 (1964) 1235

View on Pith 1964

[30] [29]

G. T. Iafrate, L. B. Mendelsohn, Phys. Rev., 182 (1969) 244

View on Pith 1969

[31] [30]

E. R. Vrscay, Phys. Rev. A, 33 (1986) 1433

View on Pith 1986

[32] [31]

F. J. Rogers, H. C. Graboske, D. J. Harwood, Phys. Rev. A, 1 (1970) 1577

View on Pith 1970

[33] [32]

J. P. Gazeau, A. Maquet, Phys. Rev. A, 20 (1979) 727

View on Pith 1979

[34] [33]

E. D. Fliho, R. M. Ricotta, Phys. Lett. A, 269 (2000) 269

View on Pith 2000

[35] [34]

B. Bagchi, C. Quesne, Phys. Lett. A, 300 (2002) 18

View on Pith 2002

[36] [35]

B. Bagchi, F. Cannata, C. Quesne, Phys. Lett. A, 269 (2000) 79. 11

View on Pith 2000

[37] [37]

C. M. Bender, S. Boettcher, Phys. Rev. Lett., 80 (1998) 5243

View on Pith 1998

[38] [38]

A. Mostafazadeh, arXiv: math-ph/0110016; ibid math-ph/0203005; ibid math- ph/0107001

View on Pith Pith review arXiv

[39] [39]

R. L. Greene, C. Aldrich, Phys. Rev. A, 14 (1976) 2363

View on Pith 1976

[40] [40]

R. Sever, C. Tezcan, Tr. J. of Physics, 17 (1993) 459-464. 12 Table 1 : Energy eigenvalues of ECSC as a function of the screening p arameter λ for 1 s, 2p, 3 d and 4 f states in Rydberg units of energy. State 1 s Screening λ SUSYQM Our Work Hypervirial Solution [40 ] NR − QM Variational [20 ] Exact Numerical [39 ] 0.020 -0.480290 -0.480310 -0.480300 -0.48...

View on Pith 1993