Ground-state energy of a particle in a space with minimal length and minimal momentum

Arsen Panas; Volodymyr Tkachuk

arxiv: 2604.27018 · v2 · submitted 2026-04-29 · 🪐 quant-ph

Ground-state energy of a particle in a space with minimal length and minimal momentum

Arsen Panas , Volodymyr Tkachuk This is my paper

Pith reviewed 2026-05-07 11:16 UTC · model grok-4.3

classification 🪐 quant-ph

keywords minimal lengthminimal momentumdeformed spaceground-state energyharmonic oscillatoruncertainty principlequantum mechanicsanharmonic oscillator

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The pith

In spaces with minimal length and minimal momentum, the ground-state energy has a rigorous lower bound that is the absolute physical minimum for a class of one-dimensional potentials.

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

The paper establishes a lower bound on the ground-state energy for quantum particles in a deformed space where both position and momentum have minimal uncertainties. This bound is derived from the deformed commutation relations and represents the lowest energy physically attainable. The authors first solve the problem for the harmonic oscillator by calculating its ground-state energy explicitly. They then generalize to a broad class of potentials by deriving an equation for the position uncertainty that minimizes the energy, which is solved numerically, and supply a linear approximation in the deformation parameters for a general expression. They also determine the range of deformation parameters where solutions exist for the anharmonic oscillator.

Core claim

In deformed spaces with minimal coordinate and momentum uncertainties, the ground-state energy for a broad class of one-dimensional potentials has a rigorous lower bound that represents the physically attainable minimum. For the harmonic oscillator this energy is calculated directly. Generalization yields an equation for the coordinate uncertainty at minimal energy, solvable numerically, with a linear approximation providing a general expression, and the domain of existence determined for anharmonic oscillators.

What carries the argument

Deformed commutation relations that enforce minimal uncertainties in both position and momentum, used to derive a variational lower bound on the ground-state energy.

If this is right

The ground-state energy of the harmonic oscillator can be calculated explicitly in this deformed space.
For general potentials the minimal energy corresponds to a specific coordinate uncertainty that solves a derived equation.
A linear approximation in the deformation parameters yields an explicit general formula for the ground-state energy.
The anharmonic oscillator has solutions only inside a restricted domain of the deformation parameters.

Where Pith is reading between the lines

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

The bound could constrain allowable values of deformation parameters when modeling real systems.
The same bounding method might extend to time-dependent or multi-dimensional potentials.
In the limit of vanishing deformation parameters the expressions recover the standard quantum-mechanical ground-state energies.

Load-bearing premise

The deformed commutation relations that enforce minimal uncertainties in both position and momentum allow a well-defined variational or operator-based lower bound to be derived that remains valid for the full class of potentials considered.

What would settle it

A numerical or experimental ground-state energy for the harmonic oscillator that lies below the derived lower bound when the deformation parameters are nonzero would disprove the central claim.

Figures

Figures reproduced from arXiv: 2604.27018 by Arsen Panas, Volodymyr Tkachuk.

**Figure 1.** Figure 1: β limit value dependence on n As can be seen from the F igure 1 β indeed holds the restriction β < 1 2 as n → ∞, as it should be. Therefore our solution does not have contradiction with exact solution for the particle in a box problem (57) in the context of the possible β values. Now it is interesting to investigate what the dependence of the existence of solutions looks like in respect to α and β. We know… view at source ↗

**Figure 2.** Figure 2: Existence regions for different values of view at source ↗

**Figure 3.** Figure 3: β limit value dependence on n for different υ0 Same goes for the domain of existence of the solution in respect to α and β, that is shrinks faster to it’s limit form as n → ∞, as it can be seen on following diagrams which plotted with fixed n = 10 and different intensity υ0. 18 view at source ↗

**Figure 4.** Figure 4: Existence regions for different values of view at source ↗

read the original abstract

In this article, we derived a rigorous lower bound on the ground-state energy for a class of one-dimensional quantum systems in deformed space with minimal coordinate and momentum uncertainties, representing the absolute minimum energy that is physically attainable. We considered a harmonic oscillator in such a space and calculate its ground-state energy. We generalized the problem to an a sufficiently broad class of potentials, deriving an equation for the coordinate uncertainty corresponding to the minimal energy, which can be solved numerically. Using a linear approximation in the deformation parameters, we obtained a general expression for the ground-state energy. We determined the domain of existence of solutions for the anharmonic oscillator potential with respect to the deformation parameters.

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

The paper gives a lower bound on ground-state energy in a space with both minimal length and minimal momentum, but the general expression rests on a linear approximation whose accuracy for finite deformations is not checked.

read the letter

The main takeaway is that this work derives a lower bound on the ground-state energy for one-dimensional systems in a deformed space that includes both minimal position and minimal momentum uncertainties. They first solve the harmonic oscillator exactly in that algebra, then set up a numerically solvable equation for the position uncertainty that minimizes the energy for a wider class of potentials, and finally extract a simple linear-in-deformation formula for the bound. They also chart the parameter region where solutions exist for the anharmonic oscillator. The numerical step and the explicit approximate formula are the concrete pieces that were not already in the literature. Adding the minimal-momentum condition to the standard minimal-length setup is incremental, but the bound itself and the domain check are useful for people who need quick estimates in these models. The soft spot is the gap between the claim of a rigorous, physically attainable lower bound and the actual derivation. The abstract presents the bound as rigorous for the broad class, yet the general expression is obtained only after a first-order truncation in the deformation parameters. No error estimate or side-by-side comparison with the full numerical minimum appears for larger deformations, so it is unclear whether higher-order terms could raise the minimum or shrink the domain of validity. The entire analysis stays in one dimension and follows the usual deformed-commutator framework without new formal tools. This paper is for readers already working on minimal-length phenomenology in quantum mechanics. Someone who needs an analytic estimate for small deformations could use the linear formula as a starting point, but would have to verify the approximation for their own parameters. I would send it for peer review. The calculations are explicit enough that referees familiar with the deformed-algebra literature can check the steps and ask for the missing error analysis.

Referee Report

1 major / 0 minor

Summary. The manuscript derives a lower bound on the ground-state energy for one-dimensional quantum systems in a deformed space with minimal uncertainties in both position and momentum, presented as the absolute minimum physically attainable energy. For the harmonic oscillator, the ground-state energy is calculated explicitly. The approach is then generalized to a broad class of potentials by deriving and numerically solving an equation for the coordinate uncertainty that minimizes the energy; a linear approximation in the deformation parameters is applied to obtain a general closed-form expression. The domain of existence of solutions is mapped for the anharmonic oscillator as a function of the deformation parameters.

Significance. If the linear approximation can be shown to preserve the lower-bound property with controlled error for finite deformations, the result would supply a practical, semi-analytic tool for estimating minimal energies in minimal-length quantum mechanics. The numerical treatment of the uncertainty equation for general potentials is a constructive element that could be extended to other deformed algebras.

major comments (1)

[Abstract and generalization to class of potentials] Abstract and the section deriving the general expression: the claim of a 'rigorous lower bound' that is 'the absolute minimum energy' is obtained only after a first-order truncation in the deformation parameters. No error bound, remainder estimate, or direct comparison between the linear result and the exact numerical minimization for finite deformation values is supplied. This truncation is load-bearing for the central assertion that the bound remains valid and physically attainable across the stated class of potentials.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need to clarify the scope of our claims regarding the lower bound. We address the major comment below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

Referee: Abstract and the section deriving the general expression: the claim of a 'rigorous lower bound' that is 'the absolute minimum energy' is obtained only after a first-order truncation in the deformation parameters. No error bound, remainder estimate, or direct comparison between the linear result and the exact numerical minimization for finite deformation values is supplied. This truncation is load-bearing for the central assertion that the bound remains valid and physically attainable across the stated class of potentials.

Authors: We agree that the abstract and the generalization section require clarification on this point. The derivation of the uncertainty equation for the coordinate variance that minimizes the energy is exact within the deformed algebra for any potential in the considered class, and the subsequent numerical solution yields the precise minimal ground-state energy for given deformation parameters. The linear approximation in the deformation parameters is applied only afterward to obtain a closed-form analytic expression. For the harmonic oscillator, the ground-state energy is computed exactly without truncation. To address the concern, we will revise the abstract to explicitly state that the general expression relies on the first-order approximation, and we will add a new subsection in the generalization section that compares the linear-approximation results against the exact numerical minimization of the uncertainty equation for finite (but small) deformation values, using the anharmonic oscillator as an example. This will include quantitative error estimates over the domain of existence of solutions. A fully rigorous remainder bound for arbitrary potentials lies beyond the present scope, but the added numerical validation will delineate the practical accuracy of the approximation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from deformed algebra

full rationale

The paper starts from the given deformed commutation relations enforcing minimal uncertainties, applies standard variational or operator-based methods to bound the ground-state energy for the harmonic oscillator, then generalizes by deriving an equation for the minimizing coordinate uncertainty (solved numerically) and applies a first-order expansion in the deformation parameters. None of these steps reduce by construction to the inputs: the uncertainty equation is obtained from energy minimization rather than being presupposed, the linear approximation is an explicit truncation with no claim that it equals the exact result, and no load-bearing self-citation or uniqueness theorem from the same authors is invoked to force the result. The bound is therefore self-contained and externally falsifiable in the zero-deformation limit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard framework of deformed quantum mechanics with minimal uncertainties; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Deformed commutation relations that enforce nonzero minimal uncertainties in both position and momentum.
Invoked to define the space in which the lower bound is derived.

pith-pipeline@v0.9.0 · 5405 in / 1198 out tokens · 59411 ms · 2026-05-07T11:16:45.530190+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Quantized Space- Time,

H. S. Snyder, “Quantized Space-Time,” Phys. Rev.71(1947), 38–41 doi:10.1103/PhysRev.71.38

work page doi:10.1103/physrev.71.38 1947
[2]

Kempf, G

A. Kempf, G. Mangano and R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys. Rev. D52(1995), 1108–1118 doi:10.1103/PhysRevD.52.1108

work page doi:10.1103/physrevd.52.1108 1995
[3]

Deformed Heisen- berg algebra and minimal length,

T. Maslowski, A. Nowicki and V. M. Tkachuk, “Deformed Heisen- berg algebra and minimal length,” J. Phys. A45(2012), 075309 doi:10.1088/1751-8113/45/7/075309 [arXiv:1201.5545 [quant-ph]]

work page doi:10.1088/1751-8113/45/7/075309 2012
[4]

Aspects of Pre-quantum De- scription of Deformed Theories,

A. M. Frydryszak and V. M. Tkachuk, “Aspects of Pre-quantum De- scription of Deformed Theories,” Czech. J. Phys.53(2003), 1035-1040 doi:10.1023/B:CJOP.0000010529.32268.03

work page doi:10.1023/b:cjop.0000010529.32268.03 2003
[5]

Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Mo- mentum and Applications to Quantum Mechanics,

C. Quesne and V. M. Tkachuk, “Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Mo- mentum and Applications to Quantum Mechanics,” SIGMA3(2007), 016 doi:10.3842/SIGMA.2007.016

work page doi:10.3842/sigma.2007.016 2007
[6]

Fonda, G

A. N. Tawfik and A. M. Diab, “A review of the generalized uncer- tainty principle,” Rep. Prog. Phys.78(2015), 126001 doi:10.1088/0034- 4885/78/12/126001

work page doi:10.1088/0034- 2015
[7]

Exact so- lution of the harmonic oscillator in arbitrary dimensions with min- imal length uncertainty relations,

L. N. Chang, D. Minic, N. Okamura and T. Takeuchi, “Exact so- lution of the harmonic oscillator in arbitrary dimensions with min- imal length uncertainty relations,” Phys. Rev. D65(2002), 125027 doi:10.1103/PhysRevD.65.125027

work page doi:10.1103/physrevd.65.125027 2002
[10]

Regularization of1/X2 potential in general case of deformed space with minimal length,

M. I. Samar and V. M. Tkachuk, “Regularization of1/X2 potential in general case of deformed space with minimal length,” J. Math. Phys.61 (2020), 092101 doi:10.1063/1.5111597

work page doi:10.1063/1.5111597 2020
[11]

Regularization of the singular inverse square potential in quantum mechanics with a minimal length,

D. Bouaziz and M. Bawin, “Regularization of the singular inverse square potential in quantum mechanics with a minimal length,” Phys. Rev. A 76(2007), 032112 doi:10.1103/PhysRevA.76.032112

work page doi:10.1103/physreva.76.032112 2007
[12]

Minimal length uncertainty relation and the hydrogen atom,

F. Brau, “Minimal length uncertainty relation and the hydrogen atom,” J. Phys. A32(1999), 7691–7696 doi:10.1088/0305-4470/32/44/308

work page doi:10.1088/0305-4470/32/44/308 1999
[13]

Perturbation hydrogen-atom spec- trum in deformed space with minimal length,

M. M. Stetsko and V. M. Tkachuk, “Perturbation hydrogen-atom spec- trum in deformed space with minimal length,” Phys. Rev. A74(2006), 012101 doi:10.1103/PhysRevA.74.012101

work page doi:10.1103/physreva.74.012101 2006
[14]

Hydrogen-atom spectrum under a minimal-length hypothesis,

S. Benczik, L. N. Chang, D. Minic and T. Takeuchi, “Hydrogen-atom spectrum under a minimal-length hypothesis,” Phys. Rev. A72(2005), 012104 doi:10.1103/PhysRevA.72.012104

work page doi:10.1103/physreva.72.012104 2005
[15]

One-dimensional Coulomb-like problem in deformed space with minimal length,

T. V. Fityo, I. O. Vakarchuk and V. M. Tkachuk, “One-dimensional Coulomb-like problem in deformed space with minimal length,” J. Phys. A39(2006), 2143–2152 doi:10.1088/0305-4470/39/9/010

work page doi:10.1088/0305-4470/39/9/010 2006
[16]

One-dimensional Coulomb-like prob- lem in general case of deformed space with minimal length,

M. I. Samar and V. M. Tkachuk, “One-dimensional Coulomb-like prob- lem in general case of deformed space with minimal length,” J. Math. Phys.57(2016), 082108 doi:10.1063/1.4961320

work page doi:10.1063/1.4961320 2016
[17]

Exact solutions for two-body problems in 1D deformed space with minimal length,

M. I. Samar and V. M. Tkachuk, “Exact solutions for two-body problems in 1D deformed space with minimal length,” J. Math. Phys.58(2017), 122108 doi:10.1063/1.4998461

work page doi:10.1063/1.4998461 2017
[18]

Regularization ofδ′(x)potential in general case of deformed space with minimal length,

M. I. Samar and V. M. Tkachuk, “Regularization ofδ′(x)potential in general case of deformed space with minimal length,” J. Phys. A55 (2022), 395303 doi:10.1088/1751-8121/ac90fe 22

work page doi:10.1088/1751-8121/ac90fe 2022
[19]

Texier and G

C. Quesne and V. M. Tkachuk, “Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework,” J. Phys. A36(2003), 10373–10391 doi:10.1088/0305- 4470/36/41/009

work page doi:10.1088/0305- 2003
[20]

Uncertainty relation in quantum mechanics with quan- tum group symmetry,

A. Kempf, “Uncertainty relation in quantum mechanics with quan- tum group symmetry,” J. Math. Phys.35(1994), 4483–4496 doi:10.1063/1.530798

work page doi:10.1063/1.530798 1994
[21]

Maximal localization in the presence of minimal uncertainties in positions and in momenta,

H. Hinrichsen and A. Kempf, “Maximal localization in the presence of minimal uncertainties in positions and in momenta,” J. Math. Phys.37 (1996), 2121–2137 doi:10.1063/1.531371

work page doi:10.1063/1.531371 1996
[22]

Quantum group symmetric Bargmann-Fock space: Integral kernels, Green functions, driving forces,

A. Kempf, “Quantum group symmetric Bargmann-Fock space: Integral kernels, Green functions, driving forces,” J. Math. Phys.34(1993), 969– 987 doi:10.1063/1.530204

work page doi:10.1063/1.530204 1993
[23]

Lower Boundary Estimation of the Ground State Energy for an Anharmonic Oscillator in Deformed Space With Minimal Length,

A. O. Panas and V. M. Tkachuk, “Lower Boundary Estimation of the Ground State Energy for an Anharmonic Oscillator in Deformed Space With Minimal Length,” J. Phys. Stud.28(2024) no.4, 4001 doi:10.30970/jps.28.4001

work page doi:10.30970/jps.28.4001 2024
[24]

New approach to nonperturbative quantum mechanics with minimal length uncertainty,

P. Pedram, “New approach to nonperturbative quantum mechanics with minimal length uncertainty,” Phys. Rev. D85(2012), 024016 doi:10.1103/PhysRevD.85.024016 23

work page doi:10.1103/physrevd.85.024016 2012

[1] [1]

Quantized Space- Time,

H. S. Snyder, “Quantized Space-Time,” Phys. Rev.71(1947), 38–41 doi:10.1103/PhysRev.71.38

work page doi:10.1103/physrev.71.38 1947

[2] [2]

Kempf, G

A. Kempf, G. Mangano and R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys. Rev. D52(1995), 1108–1118 doi:10.1103/PhysRevD.52.1108

work page doi:10.1103/physrevd.52.1108 1995

[3] [3]

Deformed Heisen- berg algebra and minimal length,

T. Maslowski, A. Nowicki and V. M. Tkachuk, “Deformed Heisen- berg algebra and minimal length,” J. Phys. A45(2012), 075309 doi:10.1088/1751-8113/45/7/075309 [arXiv:1201.5545 [quant-ph]]

work page doi:10.1088/1751-8113/45/7/075309 2012

[4] [4]

Aspects of Pre-quantum De- scription of Deformed Theories,

A. M. Frydryszak and V. M. Tkachuk, “Aspects of Pre-quantum De- scription of Deformed Theories,” Czech. J. Phys.53(2003), 1035-1040 doi:10.1023/B:CJOP.0000010529.32268.03

work page doi:10.1023/b:cjop.0000010529.32268.03 2003

[5] [5]

Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Mo- mentum and Applications to Quantum Mechanics,

C. Quesne and V. M. Tkachuk, “Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Mo- mentum and Applications to Quantum Mechanics,” SIGMA3(2007), 016 doi:10.3842/SIGMA.2007.016

work page doi:10.3842/sigma.2007.016 2007

[6] [6]

Fonda, G

A. N. Tawfik and A. M. Diab, “A review of the generalized uncer- tainty principle,” Rep. Prog. Phys.78(2015), 126001 doi:10.1088/0034- 4885/78/12/126001

work page doi:10.1088/0034- 2015

[7] [7]

Exact so- lution of the harmonic oscillator in arbitrary dimensions with min- imal length uncertainty relations,

L. N. Chang, D. Minic, N. Okamura and T. Takeuchi, “Exact so- lution of the harmonic oscillator in arbitrary dimensions with min- imal length uncertainty relations,” Phys. Rev. D65(2002), 125027 doi:10.1103/PhysRevD.65.125027

work page doi:10.1103/physrevd.65.125027 2002

[8] [10]

Regularization of1/X2 potential in general case of deformed space with minimal length,

M. I. Samar and V. M. Tkachuk, “Regularization of1/X2 potential in general case of deformed space with minimal length,” J. Math. Phys.61 (2020), 092101 doi:10.1063/1.5111597

work page doi:10.1063/1.5111597 2020

[9] [11]

Regularization of the singular inverse square potential in quantum mechanics with a minimal length,

D. Bouaziz and M. Bawin, “Regularization of the singular inverse square potential in quantum mechanics with a minimal length,” Phys. Rev. A 76(2007), 032112 doi:10.1103/PhysRevA.76.032112

work page doi:10.1103/physreva.76.032112 2007

[10] [12]

Minimal length uncertainty relation and the hydrogen atom,

F. Brau, “Minimal length uncertainty relation and the hydrogen atom,” J. Phys. A32(1999), 7691–7696 doi:10.1088/0305-4470/32/44/308

work page doi:10.1088/0305-4470/32/44/308 1999

[11] [13]

Perturbation hydrogen-atom spec- trum in deformed space with minimal length,

M. M. Stetsko and V. M. Tkachuk, “Perturbation hydrogen-atom spec- trum in deformed space with minimal length,” Phys. Rev. A74(2006), 012101 doi:10.1103/PhysRevA.74.012101

work page doi:10.1103/physreva.74.012101 2006

[12] [14]

Hydrogen-atom spectrum under a minimal-length hypothesis,

S. Benczik, L. N. Chang, D. Minic and T. Takeuchi, “Hydrogen-atom spectrum under a minimal-length hypothesis,” Phys. Rev. A72(2005), 012104 doi:10.1103/PhysRevA.72.012104

work page doi:10.1103/physreva.72.012104 2005

[13] [15]

One-dimensional Coulomb-like problem in deformed space with minimal length,

T. V. Fityo, I. O. Vakarchuk and V. M. Tkachuk, “One-dimensional Coulomb-like problem in deformed space with minimal length,” J. Phys. A39(2006), 2143–2152 doi:10.1088/0305-4470/39/9/010

work page doi:10.1088/0305-4470/39/9/010 2006

[14] [16]

One-dimensional Coulomb-like prob- lem in general case of deformed space with minimal length,

M. I. Samar and V. M. Tkachuk, “One-dimensional Coulomb-like prob- lem in general case of deformed space with minimal length,” J. Math. Phys.57(2016), 082108 doi:10.1063/1.4961320

work page doi:10.1063/1.4961320 2016

[15] [17]

Exact solutions for two-body problems in 1D deformed space with minimal length,

M. I. Samar and V. M. Tkachuk, “Exact solutions for two-body problems in 1D deformed space with minimal length,” J. Math. Phys.58(2017), 122108 doi:10.1063/1.4998461

work page doi:10.1063/1.4998461 2017

[16] [18]

Regularization ofδ′(x)potential in general case of deformed space with minimal length,

M. I. Samar and V. M. Tkachuk, “Regularization ofδ′(x)potential in general case of deformed space with minimal length,” J. Phys. A55 (2022), 395303 doi:10.1088/1751-8121/ac90fe 22

work page doi:10.1088/1751-8121/ac90fe 2022

[17] [19]

Texier and G

C. Quesne and V. M. Tkachuk, “Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework,” J. Phys. A36(2003), 10373–10391 doi:10.1088/0305- 4470/36/41/009

work page doi:10.1088/0305- 2003

[18] [20]

Uncertainty relation in quantum mechanics with quan- tum group symmetry,

A. Kempf, “Uncertainty relation in quantum mechanics with quan- tum group symmetry,” J. Math. Phys.35(1994), 4483–4496 doi:10.1063/1.530798

work page doi:10.1063/1.530798 1994

[19] [21]

Maximal localization in the presence of minimal uncertainties in positions and in momenta,

H. Hinrichsen and A. Kempf, “Maximal localization in the presence of minimal uncertainties in positions and in momenta,” J. Math. Phys.37 (1996), 2121–2137 doi:10.1063/1.531371

work page doi:10.1063/1.531371 1996

[20] [22]

Quantum group symmetric Bargmann-Fock space: Integral kernels, Green functions, driving forces,

A. Kempf, “Quantum group symmetric Bargmann-Fock space: Integral kernels, Green functions, driving forces,” J. Math. Phys.34(1993), 969– 987 doi:10.1063/1.530204

work page doi:10.1063/1.530204 1993

[21] [23]

Lower Boundary Estimation of the Ground State Energy for an Anharmonic Oscillator in Deformed Space With Minimal Length,

A. O. Panas and V. M. Tkachuk, “Lower Boundary Estimation of the Ground State Energy for an Anharmonic Oscillator in Deformed Space With Minimal Length,” J. Phys. Stud.28(2024) no.4, 4001 doi:10.30970/jps.28.4001

work page doi:10.30970/jps.28.4001 2024

[22] [24]

New approach to nonperturbative quantum mechanics with minimal length uncertainty,

P. Pedram, “New approach to nonperturbative quantum mechanics with minimal length uncertainty,” Phys. Rev. D85(2012), 024016 doi:10.1103/PhysRevD.85.024016 23

work page doi:10.1103/physrevd.85.024016 2012