Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.19611

On Solutions of the Killingbeck Potential and Clarifying Comments on a Related Analytical Approach

Fatma Zohra Khaled, Mustafa Moumni, Mokhtar Falek

quant-phSubmitted Jul 21, 2026• Announced Jul 23, 2026
arXivHTMLPDF

Abstract

The work presents analytical solutions to the Schrodinger equation for the Killingbeck potential, a Hybrid model combining harmonic, linear and Coulomb terms, and which is also an approximate model of Yukawa-type potentials. The radial Schrodinger equation is solved by means of the series expansion method, thus yielding the exact expressions of both bound-states solutions and eigen-functions for the systems; these systems include quarkonium and confined hydrogen-like atoms in plasma environments. Furthermore, we offer a constructive commentary on the work of Obu et al. (East Eur. J. Phys. 3, 146-157, 2023), with the aim of clarifying a mathematical misstatement used in their analytical treatment of analogous systems.

Journal reference
East European Journal of Physics, (4), 164-171 (2025)
DOI
10.26565/2312-4334-2025-4-14
Comments
8 pages

Related papers

Bound states of the hydrogen-like atomic systems in plasma environments

Fatma Zohra Khaled, Mustafa Moumni, Mokhtar Falek

We conduct a non-relativistic study of plasma screening effects on hydrogen-like atomic systems using the Screened Coulomb Potential (SCP i.e. Yukawa potential). The radial Schrödinger equation is first reduced to a bi-confluent Heun (BCH) equation for the Killingbeck potential which is the truncated version of the SCP, and we write the exact BCH eigenfunctions and eigenenergies. We then study the limitations of these BCH solutions and obtain an analytic description valid for weak to moderate screening using the BCH functional form. The corrected eigenfunctions are constructed order by order up to O(k^3); they are expressed in terms of the Laguerre polynomials and reduces exactly to Coulomb eigenfunctions when k=0. Using these Functions, the energy spectrum is computed via two analytic methods: (i) direct evaluation of the full Yukawa Hamiltonian expectation value, and (ii) the Hellmann-Feynman theorem, yielding integral representation. All methods presented here provide explicit analytical formulas for both wavefunctions and eigenenergies valid for different ranges of the screening. These methods establish a powerful analytic framework for studying confined quantum systems. The thermodynamic properties are also derived using the BCH formulation.

quant-pharXiv:2607.19601•Jul 23, 2026

Analytical solution of the Schrodinger equation with linear confinement potential

Hirokazu Tezuka

It is shown that the Schrodinger equation with a linear confinement potential has an analytical solution. The existence of the analytical solution requires additional terms to the potential other than the linear one. The Coulomb-type potential is necessary to get non-zero energy eigenvalues. The author proposes an asymptotically linear potential, and shows that one can analytically solve the Schrodinger equation with this potential.

Journal of Physics A Mathematical and General•Nov 21, 1991•30 citations

Bound state solutions of the Schrödinger equation for the atomic systems interacting with the radial screened Coulomb potential: analytical approximation methods

Fatma Zohra Khaled, Mustafa Moumni, Mokhtar Falek

We investigate the bound state properties of the hydrogen-like atoms in the radial screened Coulomb potential (RSCP). using three complementary analytical approaches - expectation values with Coulomb and Kratzer reference states, variational optimization with a scaled Kratzer basis, and the Hellmann-Feynman theorem - we derive approximate energy eigenvalues as function of the screening parameter c. Benchmarked against high-precision generalized pseudospectral data, the expectation value-approach with the Kratzer basis achieves relative errors of 0.63% for the first ten s-states at $c=0.1$, while the variational method improves this further. The formalism extends naturally to Positronium, demonstrating its generality for arbitrary reduced-mass systems. The complementary biases of the methods provide robust error estimation for plasma-embedded atoms.

quant-pharXiv:2607.19197•Jul 22, 2026

INFLUENCE OF EXTERNAL FIELDS ON THE KILLINGBECK POTENTIAL: QUASI EXACT SOLUTION

M. Hamzavi, Sameer M. Ikhdair

The Killingbeck potential consists of oscillator potential plus Cornell potential, i.e. ar2+ br - c/r, that it has received a great deal of attention in particle physics. In this paper, we study the energy levels and wave function for arbitrary m-state in two-dimensional (2D) Schrödinger equation (SE) with a Killingbeck potential under the influence of strong external uniform magnetic and Aharonov–Bohm (AB) flux fields perpendicular to the plane where the interacting particles are confined. We use the wave function ansatz method to solve the radial problem of the Schrödinger equation with Killingbeck potential. We obtain the energy levels in the absence of external fields and also find the energy levels of the familiar Coulomb and harmonic oscillator potentials.

Modern Physics Letters B•Sep 11, 2013•6 citations

A method for solving the Schrodinger equation

Nathan Poliatzky

The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.

Journal of Physics A Mathematical and General•Jun 21, 1992•7 citations

Factorization method for exact solution of the non-central modified Killingbeck potential plus a ring-shaped-like potential

B. Tchana Mbadjoun, J. M. Ema’a Ema’a, Jean Yomi +3 more

In this paper, we study the Schrödinger equation with non-central modified Killingbeck potential plus a ring-shaped-like potential problem, which is not spherically symmetric. The factorization method is used to solve the hypergeometric equation types which lead to solutions with the associate Laguerre function for the radial part and Jacobi polynomial for the polar part. We introduce the raising and lowering operators to calculate the energies eigenvalues, which show that the lack of spherical symmetry removes the degeneracy of second quantum number m which is completely expected. These obtained energies are better to explain the superposition of the energy levels of the atoms in the crystalline structure of molecules.

Modern Physics Letters A•Mar 12, 2019•9 citations
The work presents analytical solutions to the Schrodinger equation for the Killingbeck potential, a Hybrid model combining harmonic, linear and Coulomb terms, and which is also an approximate model of Yukawa-type potentials. The radial Schrodinger equation is solved by means of the series expansion method, thus yielding the exact expressions of both bound-states solutions and eigen-functions for the systems; these systems include quarkonium and confined hydrogen-like atoms in plasma environments. Furthermore, we offer a constructive commentary on the work of Obu et al. (East Eur. J. Phys. 3, 146-157, 2023), with the aim of clarifying a mathematical misstatement used in their analytical treatment of analogous systems.
We conduct a non-relativistic study of plasma screening effects on hydrogen-like atomic systems using the Screened Coulomb Potential (SCP i.e. Yukawa potential). The radial Schrödinger equation is first reduced to a bi-confluent Heun (BCH) equation for the Killingbeck potential which is the truncated version of the SCP, and we write the exact BCH eigenfunctions and eigenenergies. We then study the limitations of these BCH solutions and obtain an analytic description valid for weak to moderate screening using the BCH functional form. The corrected eigenfunctions are constructed order by order up to O(k^3); they are expressed in terms of the Laguerre polynomials and reduces exactly to Coulomb eigenfunctions when k=0. Using these Functions, the energy spectrum is computed via two analytic methods: (i) direct evaluation of the full Yukawa Hamiltonian expectation value, and (ii) the Hellmann-Feynman theorem, yielding integral representation. All methods presented here provide explicit analytical formulas for both wavefunctions and eigenenergies valid for different ranges of the screening. These methods establish a powerful analytic framework for studying confined quantum systems. The thermodynamic properties are also derived using the BCH formulation.
We investigate the bound state properties of the hydrogen-like atoms in the radial screened Coulomb potential (RSCP). using three complementary analytical approaches - expectation values with Coulomb and Kratzer reference states, variational optimization with a scaled Kratzer basis, and the Hellmann-Feynman theorem - we derive approximate energy eigenvalues as function of the screening parameter c. Benchmarked against high-precision generalized pseudospectral data, the expectation value-approach with the Kratzer basis achieves relative errors of 0.63% for the first ten s-states at c=0.1c=0.1c=0.1, while the variational method improves this further. The formalism extends naturally to Positronium, demonstrating its generality for arbitrary reduced-mass systems. The complementary biases of the methods provide robust error estimation for plasma-embedded atoms.
The Killingbeck potential consists of oscillator potential plus Cornell potential, i.e. ar2+ br - c/r, that it has received a great deal of attention in particle physics. In this paper, we study the energy levels and wave function for arbitrary m-state in two-dimensional (2D) Schrödinger equation (SE) with a Killingbeck potential under the influence of strong external uniform magnetic and Aharonov–Bohm (AB) flux fields perpendicular to the plane where the interacting particles are confined. We use the wave function ansatz method to solve the radial problem of the Schrödinger equation with Killingbeck potential. We obtain the energy levels in the absence of external fields and also find the energy levels of the familiar Coulomb and harmonic oscillator potentials.
The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.
In this paper, we study the Schrödinger equation with non-central modified Killingbeck potential plus a ring-shaped-like potential problem, which is not spherically symmetric. The factorization method is used to solve the hypergeometric equation types which lead to solutions with the associate Laguerre function for the radial part and Jacobi polynomial for the polar part. We introduce the raising and lowering operators to calculate the energies eigenvalues, which show that the lack of spherical symmetry removes the degeneracy of second quantum number m which is completely expected. These obtained energies are better to explain the superposition of the energy levels of the atoms in the crystalline structure of molecules.