Hakan Ciftci, Richard L. Hall, Nasser Saad
In a previous paper (J. Phys. A 36, 11807 (2003)), we introduced the `asymptotic iteration method' for solving second-order homogeneous linear differential equations. In this paper, we study perturbed problems in quantum mechanics and we use the method to find the coefficients in the perturbation series for the eigenvalues and eigenfunctions directly, without first solving the unperturbed problem.
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We have used Asymptotic Iteration Method (AIM) for obtaining the eigenvalues of the Schrodinger's equation for a deformed well problem representing trigonometric functions. By solving the problem, we have found that the Schrodinger's equation for the considered potential has quasi-exact solutions. Additionally, we have also calculated the perturbation expansion of energy eigenvalues and found very simple analytical expression of the energy. Finally, we have considered more general cases and obtained energy eigenvalues for arbitrary potential parameters.
This paper features and elaborates recent developments and modifications in asymptotic techniques in solving differential equation in non linear dynamics. These methods are proved to be powerful to solve weakly as well as strongly non linear cases. Obtained approximate analytical solutions are valid for the whole solution domain. In this paper, limitations of traditional perturbation methods are illustrated with various modified techniques. Mathematical tools such as variational approach, homotopy and iteration technique are discussed to solve various problems efficiently. Asymptotic methods s…
A simple exact analytical solution of the relativistic Duffin-Kemmer-Petiau equation within the framework of the asymptotic iteration method is presented. Exact bound state energy eigenvalues and corresponding eigenfunctions are determined for the relativistic harmonic oscillator as well as the Coulomb potentials. As a non-trivial example, the anharmonic oscillator is solved and the energy eigenvalues are obtained within the perturbation theory using the asymptotic iteration method.
We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue problems which includes Schroedinger problems with Coulomb, harmonic oscillator, or Poeschl-Teller potentials, as well as the special eigenproblems studied recently by Bender et al [J. Phys. A: Math. Gen. 34 9835 (2001)] and generalized in the present paper to higher dimensions.
We show that recent applications of the homotopy perturbation method the Adomian decomposition method and the variational iteration method are completely useless for the treatment of nonlinear problems.
For non-zero $\ell$ values, we present an analytical solution of the radial Schrödinger equation for the rotating Morse potential using the Pekeris approximation within the framework of the Asymptotic Iteration Method. The bound state energy eigenvalues and corresponding wave functions are obtained for a number of diatomic molecules and the results are compared with the findings of the super-symmetry, the hypervirial perturbation, the Nikiforov-Uvarov, the variational, the shifted 1/N and the modified shifted 1/N expansion methods.
The Asymptotic Iteration Method (AIM) is a technique for solving analytically and approximately the linear second-order differential equation, especially the eigenvalue problems that frequently appear in theoretical and mathematical physics. The analysis and mathematical justifications of the success and failure of the asymptotic iteration method are detailed in this work. A theorem explaining why the asymptotic iteration method works for the eigenvalue problem is presented. As a byproduct, a new procedure to generate unlimited classes of exactly solvable differential equations is also introdu…
Many problems in physics, chemistry and other fields are perturbative in nature, i.e. differ only slightly from related problems with known solutions. Prominent among these is the eigenvalue perturbation problem, wherein one seeks the eigenvectors and eigenvalues of a matrix with small off-diagonal elements. Here we introduce a novel iterative algorithm to compute these eigenpairs based on fixed-point iteration for an algebraic equation in complex projective space. We show from explicit and numerical examples that our algorithm outperforms the usual Rayleigh-Schrödinger expansion on three coun…
The asymptotic iteration method is used to find exact and approximate solutions of Schroedinger'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 Schroedinger 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.
In this article, we present the analytical solution of the radial Schrödinger equation for the Hulthén potential within the framework of the asymptotic iteration method by using an approximation to the centrifugal potential for any $l$ states. We obtain the energy eigenvalues and the corresponding eigenfunctions for different screening parameters. The wave functions are physical and energy eigenvalues are in good agreement with the results obtained by other methods for different $δ$ values. In order to demonstrate this, the results of the asymptotic iteration method are compared with the resul…
The purpose of this comment is to present the perturbation approach proposed by Maiz J. Math. Phys. 60, 012103 (2019) in a clearer way. The results of our straightforward procedure agree with those obtained by that author except for one case in which we obtain the exact result while he obtained an approximate one. In addition to it, we show that for sufficiently deep double-well potentials the perturbation approach deteriorates considerably.
New probabilistic methods for physics
本文提出了一种新的概率方法，用于描述迭代的渐近行为，并应用于常微分方程和力学问题的逼近。
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