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Exact and approximate solutions to Schrödinger’s equation with decatic potentials

100%
Brandon D., Saad N.
Open Physics
|
2013
|
vol. 11
|
issue 3
279-290
EN
The one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax 10 + bx 8 + cx 6 + dx 4 + ex 2, a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.
2
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Schrödinger spectrum generated by the Cornell potential

100%
Hall R. L., Saad N.
Open Physics
|
2015
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vol. 13
|
issue 1
EN
The eigenvalues Ednl (a, c) of the d-dimensional Schrödinger equation with the Cornell potential V(r) = −a/r + c r, a, c > 0 are analyzed by means of the envelope method and the asymptotic iteration method (AIM). Scaling arguments show that it is suffcient to know E(1, λ), and the envelope method provides analytic bounds for the equivalent complete set of coupling functions λ(E). Meanwhile the easily-implemented AIM procedure yields highly accurate numerical eigenvalues with little computational effort.
3
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The Klein-Gordon equation with the Kratzer potential in d dimensions

81%
Saad N., Hall R., Ciftci H.
Open Physics
|
2008
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vol. 6
|
issue 3
717-729
EN
We apply the Asymptotic Iteration Method to obtain the bound-state energy spectrum for the d-dimensional Klein-Gordon equation with scalar S(r) and vector potentials V(r). When S(r) and V(r) are both Coulombic, we obtain all the exact solutions; when the potentials are both of Kratzer type, we obtain all the exact solutions for S(r) = V(r); if S(r) > V(r) we obtain exact solutions under certain constraints on the potential parameters: in this case, a possible general solution is found in terms of a monic polynomial, whose coefficients form a set of elementary symmetric polynomials.
4
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Exact and approximate solutions of Schrödinger’s equation for a class of trigonometric potentials

81%
Ciftci H., Hall R., Saad N.
Open Physics
|
2013
|
vol. 11
|
issue 1
37-48
EN
The asymptotic iteration method is used to find exact and approximate solutions of Schrödinger’s equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent). Analytic and approximate solutions are obtained by first using a coordinate transformation to reduce the Schrödinger equation to a second-order differential equation with an appropriate form. The asymptotic iteration method is also employed indirectly to obtain the terms in perturbation expansions, both for the energies and for the corresponding eigenfunctions.
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