Closed-form solutions for the Schrödinger wave equation with non-solvable potentials: A perturbation approach

Abstract To obtain closed-form solutions for the radial Schrödinger wave equation with non-solvable potential models, we use a simple, easy, and fast perturbation technique within the framework of the asymptotic iteration method (PAIM). We will show how the PAIM can be applied directly to find the analytical coefficients in the perturbation series, without using the base eigenfunctions of the unperturbed problem. As an example, the vector Coulomb ( ∼ 1 / r ) \left( \sim 1\hspace{0.1em}\text{/}\hspace{0.1em}r) and the harmonic oscillator ( ∼ r 2 ) \left( \sim {r}^{2}) plus linear ( ∼ r ) \left( \sim r) scalar potential parts implemented with their counterpart spin-dependent terms are chosen to investigate the meson sectors including charm and beauty quarks. This approach is applicable in the same form to both the ground state and the excited bound states and can be easily applied to other strongly non-solvable potential problems. The procedure of this method and its results will provide a valuable hint for investigating tetraquark configuration.

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Closed-form solutions to the solitary wave equation in an unmagnatized dusty plasma

Alexandria Engineering Journal ◽

10.1016/j.aej.2020.03.030 ◽

2020 ◽

Vol 59 (3) ◽

pp. 1505-1514 ◽

Cited By ~ 2

Author(s):

Md Nur Alam ◽

Aly R. Seadawy ◽

Dumitru Baleanu

Keyword(s):

Wave Equation ◽

Solitary Wave ◽

Closed Form ◽

Dusty Plasma ◽

Closed Form Solutions

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THE ASYMPTOTIC ITERATION METHOD FOR DIRAC AND KLEIN–GORDON EQUATIONS WITH A LINEAR SCALAR POTENTIAL

International Journal of Modern Physics A ◽

10.1142/s0217751x06030916 ◽

2006 ◽

Vol 21 (19n20) ◽

pp. 4127-4135 ◽

Cited By ~ 31

Author(s):

T. BARAKAT

Keyword(s):

Iteration Method ◽

Bound States ◽

Scalar Potential ◽

Asymptotic Iteration Method ◽

Klein Gordon ◽

Linear Scalar

The asymptotic iteration method is used for Dirac and Klein–Gordon equations with a linear scalar potential to obtain the relativistic eigenenergies. A parameter, ς = 0, 1, is introduced in such a way that one can obtain Klein–Gordon bound states from Dirac bound states. It is shown that this method asymptotically gives accurate results for both Dirac and Klein–Gordon equations.

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On closed-form solutions of some nonlinear partial differential equations

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171200001289 ◽

2000 ◽

Vol 23 (2) ◽

pp. 81-88 ◽

Cited By ~ 1

Author(s):

S. S. Okoya

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Wave Equation ◽

Steady State ◽

Closed Form ◽

Nonlinear Wave Equation ◽

Nonlinear Partial Differential Equations ◽

Partial Differential ◽

Closed Form Solutions ◽

Thermal Explosion Theory

This paper is devoted to closed-form solutions of the partial differential equation:θxx+θyy+δexp(θ)=0, which arises in the steady state thermal explosion theory. We find simple exact solutions of the formθ(x,y)=Φ(F(x)+G(y)), andθ(x,y)=Φ(f(x+y)+g(x-y)). Also, we study the corresponding nonlinear wave equation.

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A point source in the presence of spherical material inhomogenity: Analysis of two approximate closed form solutions for electrical scalar potential

Facta universitatis - series Electronics and Energetics ◽

10.2298/fuee0903265r ◽

2009 ◽

Vol 22 (3) ◽

pp. 265-284 ◽

Cited By ~ 1

Author(s):

Predrag Rancic ◽

Miodrag Stojanovic ◽

Milica Rancic ◽

Nenad Cvetkovic

Keyword(s):

Closed Form ◽

Numerical Experiments ◽

Scalar Potential ◽

Low Frequency ◽

Approximate Solutions ◽

Green Functions ◽

Closed Form Solutions ◽

Very Low Frequency ◽

Periodic Current ◽

Approximate Expressions

A brief review of derivation of two groups of approximate closed form expressions for the electrical scalar potential (ESP) Green functions that originates from the current of the point ground electrode (PGE) in the presence of a spherical ground inhomogenity, are presented in this paper. The PGE is fed by a very low frequency periodic current through a thin isolated conductor. One of approximate solutions is proposed in this paper. Known exact solutions that have parts in a form of infinite series sums are also given in this paper. Here, the exact solution is solely reorganized in order to facilitate comparison to the closed form solutions, and to estimate the error introduced by the approximate solutions. Finally, error estimation is performed comparing the results for the electrical scalar potential obtained applying the approximate expressions and accurate calculations. This is illustrated by numerous numerical experiments. .

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Optimal Perturbation Technique within the Asymptotic Iteration Method for Heavy-Light Meson Mass Splittings

Universe ◽

10.3390/universe7060180 ◽

2021 ◽

Vol 7 (6) ◽

pp. 180

Author(s):

Haifa I. Alrebdi ◽

Thabit Barakat

Keyword(s):

Iteration Method ◽

Perturbation Technique ◽

Light Meson ◽

Asymptotic Iteration Method ◽

Meson Mass ◽

Relativistic Wave Equation ◽

Expectation Values ◽

Optimal Perturbation ◽

Dirac Delta ◽

Unperturbed Problem

For further insight into the perturbation technique within the framework of the asymptotic iteration method (PAIM), we suggest this method to be used as an alternative method to the traditional well-known perturbation techniques. We show by means of very simple algebraic manipulations that PAIM can be directly applied to obtain the symbolic expectation value of any perturbed potential piece without using the eigenfunction of the unperturbed problem. One of the fundamental advantages of PAIM is its ability to extract a reference unperturbed potential piece or pieces from the total Hamiltonian which can be solved exactly within AIM. After all, one can easily compute the symbolic expectation values of the remaining potential pieces. As an example, the present scheme is applied to the semi-relativistic wave equation with the harmonic-oscillator potential implemented with the Fermi–Breit potential terms. In particular, the non-trivial symbolic expectation values of the Dirac delta function, and the momentum-dependent orbit–orbit coupling terms are successfully calculated. Results are then used, as an illustration, to compute the semi-relativistic s-wave heavy-light meson masses. We obtain good agreement with experimental data for the meson mass splittings cu¯, cd¯, cs¯, bu¯, bd¯, bs¯.

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