A set of nonlinear coherent states for the pseudoharmonic oscillator

Abdelhak Kouraich; Hamidou Kassogue; Khalid Ahbli; Patrick K. Kayupe

arxiv: 1907.08881 · v1 · pith:PTJTKE4Anew · submitted 2019-07-20 · 🧮 math-ph · math.MP

A set of nonlinear coherent states for the pseudoharmonic oscillator

Khalid Ahbli , Hamidou Kassogue , Patrick K. Kayupe , Abdelhak Kouraich This is my paper

Pith reviewed 2026-05-24 18:24 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords nonlinear coherent statespseudoharmonic oscillatorgeneralized factorialresolution of identityBarut-Girardello statesBargmann transform

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The pith

A two-parameter family of nonlinear coherent states is built for the pseudoharmonic oscillator by replacing n! with a generalized factorial in the expansion coefficients.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper replaces the factorial in the standard coherent-state series z^n / sqrt(n!) with a specific generalized factorial x_n^{γ,σ}! and selects the parameters γ and σ so that the resulting states remain normalized and obey the resolution-of-identity condition. This produces a family that includes the Barut-Girardello states and the philophase states as special cases. When γ and σ take particular values the states become superpositions of the energy eigenstates of the two-parameter pseudoharmonic oscillator. The authors also introduce the associated Bargmann-type transform and record several of its properties.

Core claim

The central claim is that a two-parameter family of nonlinear coherent states exists, obtained by substituting the generalized factorial x_n^{γ,σ}! for n! in the canonical coherent-state coefficients; the values of γ and σ are chosen so that both the normalization sum and the integral that expresses resolution of the identity are satisfied when the states are attached to the eigenstates of the pseudoharmonic oscillator with parameters α, β > 0.

What carries the argument

The generalized factorial x_n^{γ,σ}! that replaces n! in the coherent-state coefficients, together with the conditions on γ and σ that guarantee convergence of the normalization sum and the resolution-of-identity integral.

If this is right

The constructed states form an overcomplete set for the pseudoharmonic oscillator when the parameter conditions are met.
The states reduce to the Barut-Girardello coherent states for one limiting choice of γ and σ.
The Bargmann-type transform maps the nonlinear coherent states to holomorphic functions whose inner product reproduces the resolution integral.
Expectation values of observables in these states can be computed directly from the analytic representation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same substitution rule may be applied to other potentials whose spectrum permits a suitable generalized factorial.
Time-dependent states built from these nonlinear coherent states would evolve under the pseudoharmonic Hamiltonian in a manner controlled by the analytic function in the Bargmann space.
The resolution-of-identity property supplies a new resolution of the identity that could be used to define a phase-space representation for the oscillator.

Load-bearing premise

There exist real numbers γ and σ for which the sum that defines the normalization constant converges and the integral that expresses resolution of the identity equals the identity operator on the pseudoharmonic-oscillator Hilbert space.

What would settle it

An explicit calculation showing that, for every pair γ, σ that satisfies the preliminary convergence conditions, either the normalization sum diverges or the resolution integral fails to reproduce the identity operator.

read the original abstract

We construct two-parameters family of nonlinear coherent states by replacing the factorial in coefficients $z^n/\sqrt{n!}$ of the canonical coherent states by a specific generalized factorial $x_n^{\gamma,\sigma}!$ where parameters $\gamma$ and $\sigma$ satisfy some conditions for which the normalization condition and the resolution of identity are verified. The obtained family is a generalization of the Barut-Girardello coherent states and those of the philophase states. In the particular case of parameters $\gamma$ and $\sigma$, we attache these states to the pseudo-harmonic oscillator depending on two parameters $\alpha,\beta> 0$. The obtained nonlinear coherent states are superposition of eigenstates of this oscillator. The associated Bargmann-type transform is defined and we derive some results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper adds a two-parameter nonlinear coherent state family for the pseudoharmonic oscillator via a custom generalized factorial, extending prior constructions in a standard way.

read the letter

The paper builds a two-parameter family of nonlinear coherent states by replacing n! with a generalized factorial x_n^{γ,σ}! in the usual expansion, then ties the states to the eigenbasis of the pseudoharmonic oscillator with parameters α, β > 0. It also defines an associated Bargmann-type transform and derives some results from it. This is a direct extension of the Barut-Girardello and philophase states mentioned in the abstract, with the new element being the specific attachment to this oscillator and the two free parameters γ, σ that are supposed to make normalization and resolution of the identity hold. The construction itself follows the usual replacement technique in the nonlinear coherent states literature, so the advance is incremental but concrete. The states are written explicitly as superpositions over the oscillator eigenstates, which is useful for anyone who wants to compute overlaps or expectation values in this model. The soft spot is the verification of the parameter conditions. The abstract states that suitable γ and σ exist so both the normalization sum and the resolution-of-identity integral work, but if the paper only asserts this without explicit bounds, convergence checks, or a clear derivation for the oscillator spectrum, that step remains the least secure part. The stress-test concern about whether the generalized factorial preserves a positive measure without extra domain restrictions on z lands here; a reader would want to see the actual calculation rather than the claim. This is for specialists in coherent states for specific potentials or generalized factorials in mathematical physics. A reader already working in that niche would find the explicit family and transform worth looking at. It deserves peer review because the construction is well-defined and the claims are in principle falsifiable, even if the verification section may need tightening.

Referee Report

2 major / 2 minor

Summary. The paper constructs a two-parameter family of nonlinear coherent states by replacing n! in the canonical coherent-state expansion z^n / √n! with a generalized factorial x_n^{γ,σ}!, where parameters γ and σ are chosen so that the resulting states are normalized and satisfy the resolution of the identity. The family generalizes the Barut-Girardello and philophase coherent states; in a special case the states are attached to the eigenbasis of the pseudoharmonic oscillator with parameters α, β > 0, yielding superpositions |z⟩ = N(|z|) ∑ (z^n / √(x_n^{γ,σ}!)) |n; α, β⟩, and a Bargmann-type transform is defined.

Significance. If the existence of admissible (γ, σ) is shown explicitly for the concrete spectrum of the pseudoharmonic oscillator, the construction supplies a concrete, parameter-controlled deformation of coherent states that preserves both normalization and over-completeness, extending known families to a solvable potential of physical interest.

major comments (2)

[attachment section (with α, β)] Abstract and the section on attachment to the pseudoharmonic oscillator: the central claim that “parameters γ and σ satisfy some conditions for which the normalization condition and the resolution of identity are verified” is asserted without an explicit derivation, error estimate, or verification that the same (γ, σ) simultaneously make both the sum N(|z|)^{-2} = ∑ |z|^{2n} / x_n^{γ,σ}! convergent and positive and the integral ∫ |z⟩⟨z| dμ(z) = I hold when the |n; α, β⟩ are the actual eigenstates of the oscillator. This verification is load-bearing for the attachment claim.
[definition of x_n^{γ,σ}!] Definition of the generalized factorial x_n^{γ,σ}! and the subsequent normalization sum: the manuscript must exhibit the concrete recurrence or closed form used for x_n^{γ,σ}! and demonstrate that the radius of convergence of the series for N(|z|) remains positive for the chosen (γ, σ) when the energy eigenvalues depend on α and β; otherwise the domain of z on which the states are defined may be empty or trivial.

minor comments (2)

Notation for the measure dμ(z) should be introduced once and used consistently; the current text alternates between “resolution of identity” and explicit integral without a single defining equation.
The statement that the family “is a generalization of the Barut-Girardello coherent states and those of the philophase states” would benefit from a short table or paragraph comparing the limiting cases of (γ, σ) to the known constructions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

Referee: Abstract and the section on attachment to the pseudoharmonic oscillator: the central claim that parameters γ and σ satisfy some conditions for which the normalization condition and the resolution of identity are verified is asserted without an explicit derivation, error estimate, or verification that the same (γ, σ) simultaneously make both the sum N(|z|)^{-2} convergent and positive and the integral hold when the |n; α, β⟩ are the actual eigenstates.

Authors: We agree that the verification for the attachment to the pseudoharmonic oscillator eigenstates must be made explicit. In the revised manuscript we will add a dedicated derivation of the admissible (γ, σ) pairs that simultaneously guarantee convergence and positivity of the normalization sum together with the resolution of the identity for the concrete spectrum depending on α, β > 0, including the explicit form of the measure dμ(z). revision: yes
Referee: Definition of the generalized factorial x_n^{γ,σ}! and the subsequent normalization sum: the manuscript must exhibit the concrete recurrence or closed form used for x_n^{γ,σ}! and demonstrate that the radius of convergence of the series for N(|z|) remains positive for the chosen (γ, σ) when the energy eigenvalues depend on α and β.

Authors: We accept that the definition and convergence analysis require explicit presentation. The revised version will state the recurrence (or closed form) for x_n^{γ,σ}! and prove that the radius of convergence of the normalization series stays positive for the selected (γ, σ) independently of the α, β-dependent energies, thereby ensuring a non-empty domain for z. revision: yes

Circularity Check

0 steps flagged

No circularity: construction proceeds by direct definition and parameter selection

full rationale

The paper defines nonlinear coherent states by substituting a two-parameter generalized factorial into the canonical coherent-state expansion and then selects γ, σ so that the resulting normalization sum converges and a positive measure exists for the resolution of the identity. This is an explicit constructive procedure whose validity is verified by direct computation on the chosen basis, not by any reduction of the claimed states to a fitted quantity or to a prior result whose only justification is self-citation. No load-bearing uniqueness theorem, ansatz smuggled via citation, or renaming of an empirical pattern appears in the derivation chain; the attachment to the pseudoharmonic oscillator is likewise a straightforward superposition in the given eigenbasis. The central claim therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The construction rests on the existence of a generalized factorial whose parameters can be tuned to satisfy two analytic conditions; no new physical entities are introduced.

free parameters (2)

γ
One of the two free parameters in the generalized factorial; its admissible range is chosen so that normalization and resolution of identity hold.
σ
Second free parameter in the generalized factorial; admissible range chosen to satisfy the same analytic conditions.

axioms (2)

domain assumption The series defining the generalized factorial x_n^{γ,σ}! converges for the chosen γ, σ and produces positive real numbers suitable for a coherent-state expansion.
Invoked when the replacement z^n / √n! → z^n / √(x_n^{γ,σ}!) is performed and normalization is asserted.
standard math The eigenstates of the pseudoharmonic oscillator form a complete orthonormal basis that can be used to verify the resolution of the identity integral.
Standard assumption in the theory of coherent states for a given Hamiltonian; invoked when the states are attached to the oscillator.

pith-pipeline@v0.9.0 · 5670 in / 1546 out tokens · 22091 ms · 2026-05-24T18:24:50.727536+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 2 internal anchors

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Schr¨ odinger, Der stetige ¨ ubergang von der mikro-zur makromechanik

E. Schr¨ odinger, Der stetige ¨ ubergang von der mikro-zur makromechanik. Naturwissenschaften, 14 (1926) 664-666

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Shanta, S

P. Shanta, S. Chaturvdi, V. Srinivasan and R. Jagannathan, Un iﬁed approach to the analogues of single- photon and multiphoton coherent states for generalized bosonic o scillators, J. Phys. A: Math. Gen. 27 (1994) 6433

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R. L. de Matos Filho and W. Vogel, Nonlinear coherent states, Phys. Rev. A 54 (1996) 4560

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V. I. Man’ko, G. Marmo, E. C. G. Sudarshan and F. Zaccaria, f-o scillators and non-linear coherent states, Phys. Scr. 55 (1997) 528

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Roknizadeh and M

R. Roknizadeh and M. K. Tavassoly, The construction of some imp ortant classes of generalized coherent states: the nonlinear coherent states method, J. Phys. A: Math. Gen. 37 (2004) 8111 16 A SET OF NONLINEAR COHERENT STATES FOR THE PSEUDOHARMONIC OSCILLATOR

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A. O. Barut and L. Girardello, New coherent states associated w ith Non-compact groups, commun. mat. phys. 21, 1971

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Brif, Photon states associated with Holstein-Primakoﬀ realiza tion of SU (1, 1) Lie algebra, Quantum Semiclass

C. Brif, Photon states associated with Holstein-Primakoﬀ realiza tion of SU (1, 1) Lie algebra, Quantum Semiclass. Opt. 7 803, 1995

work page 1995

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Ahbli, P

K. Ahbli, P. Kayupe Kikiodio and Z. Mouayn, Orthogonal polynomials attached to coherent states for the symmetric P¨ oschl-Teller oscillator.Integral Transforms Spec Funct. 2016; 27(10):806-823

work page 2016

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S. T. Ali and M. E. H. Ismail, Some orthogonal polynomials arising fr om coherent states, J.Phys A: Math. Theor, 45 (2012)

work page 2012

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I. I. Gol’dman and D. V. Krivchenkov, Problems in Quantum Mecha nics (London: Pergamon; 1961)

work page 1961

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Mojaveri, A

B. Mojaveri, A. Dehghani and R. J. Bahrbeig, Nonlinear cohere nt states of the para-Bose oscillator and their non-classical features, Eur. Phys. J. Plus , (2018) 133 : 529

work page 2018

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Coherent States, Wavelets, and their Generalizations (New york: Springer Science + Busness Media ; 1999, 2014)

Ali ST, Antoine JP and Gazeau JP. Coherent States, Wavelets, and their Generalizations (New york: Springer Science + Busness Media ; 1999, 2014)

work page 1999

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Iwata, Non-Hermitian operators and eigenfunction expans ions Prog

G. Iwata, Non-Hermitian operators and eigenfunction expans ions Prog. Theor. Phys. 6, 1951

work page 1951

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G. N. Watson, A treatise on the theory of Bessel Functions (C ambridge: Sc. D., F. R. S; 1944)

work page 1944

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H. M. Srivastava and H. L. Manocha, A Treatise on Generating F unctions (London: Ellis Horwood Limited; 1984)

work page 1984

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Appl and D

T. Appl and D. H. Schiller, Generalized hypergeometric coheren t states, J. Phys. A: Math. Gen. 37 (2004) 2731

work page 2004

[17] [17]

Popov and M

D. Popov and M. Popov, Some operational properties of the ge neralized hypergeometric coherent states, Phys. Scr. 90 (2015) 035101

work page 2015

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Erdelyi et al., Tables of Integral Transforms, vol.1, (New Yo rk: McGraw-Hill; 1954)

A. Erdelyi et al., Tables of Integral Transforms, vol.1, (New Yo rk: McGraw-Hill; 1954)

work page 1954

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Sixdeniers, and K.A

J.M. Sixdeniers, and K.A. Penson. On the completeness of coher ent states generated by binomial dis- tribution. J. Phys. A: Math. Gen. 33 (2000) 2907-2916

work page 2000

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O. I. Marichev, Handbook of Integral Transforms of Higher T ranscendental Functions: Theory and Algorithmic Tables (Chichester: Ellis Horwood; 1983)

work page 1983

[21] [21]

A. M. Mathai and R. K. Saxena , Genaralized Hypergeometric fu nction with application in statistics and physical sciences (Heidelberg: Springer-Verlag Berlin; 1973)

work page 1973

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Oberhettinger, Tables of Mellin transforms (Hildelberg: Sprin ger-Verlag Berlin; 1974)

F. Oberhettinger, Tables of Mellin transforms (Hildelberg: Sprin ger-Verlag Berlin; 1974)

work page 1974

[23] [23]

R. L. Hall, N. Saad and A. B. Von Keviczky, Closed-form sums for some perturbation series involving associated Laguerre polynomials J. Phys. A: Math. Gen. 34 (2001) 11287-300

work page 2001

[24] [24]

R. L. Hall, N. Saad and A. B. Von Keviczky, Spiked harmonic oscillat ors arXiv:math-ph/0109014v1, 2001

work page internal anchor Pith review Pith/arXiv arXiv 2001

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S. M. Nagiev, E. I. Jafarov and R. M. Imanov, On a dynamical sy mmetry of the relativistic linear singular oscillator, arXiv:math-ph/0608057v2, 2006

work page internal anchor Pith review Pith/arXiv arXiv 2006

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Magnus, F

W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theo rems for the Special Functions of Mathematical Physics (Berlin: Springer; 1966)

work page 1966

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I. S. Gradshteyn and I. M. Ryzhik, Table of integrales, series a nd products (Elsevier Inc.; 2007) E-mail address : ⋆ ahbli.khalid@gmail.com, ♭hamidoukass@gmail.com, ‡Kayupepatrick@gmail.com, Υakouraich@yahoo.fr

work page 2007