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Perturbation theory in the framework of the improved asymptotic iteration method

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Abstract

Based on the concepts of the improved asymptotic iteration method, we modify the original asymptotic iteration method for perturbation problems previously published. In our computations, the proposed method proves to be significantly faster than the previous perturbative method as we make explicit in the examples given in this work. Our procedure enables us to compute the corrections to the energy eigenvalues to an order as high as 15 in numeric computations, and of order as high as 5 in symbolic ones.

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Data Availability Statement

A Mathematica notebook generated during the current study is available from the corresponding author on reasonable request. The manuscript has associated data in a data repository

Notes

  1. This condition is also named as: quantization condition or termination condition [4, 5, 8].

  2. A Mathematica notebook that generates the data of Table 1 is available as online supplementary material. Also, this notebook is available from the corresponding author.

  3. For the complex cubic anharmonic oscillator, from the numerical results of Table 5, we infer that in the analytical expression \(E_n^{(4)}\) given in Ref. [3] a global minus sign is missing (see also Table 6).

  4. For the perturbed Pöschl–Teller potential we calculated the analytical expression for the corrections of order \(i=5\), but for simplicity we do not show this in the tables because of the size of the expression. If the readers are interested, they can email us and we can gladly share the expression.

  5. We have noted an error in Eq. (34) of Ref. [3] for the expression they call \(u_0\). We further investigated this point and we concluded that this expression must have a global minus sign.

  6. We noted that our analytical expressions are valid starting from \(n=0\) as those of Ref. [6], although they comment in a few places that some of their expressions are valid only for \(n=1,2,3,\ldots\).

  7. In this section, we follow Ref. [7], and therefore, the Schrödinger equation has an additional factor of (1/2) multiplying the second derivative, compared with the conventions used in the previous sections, but in contrast to Ref. [7], we denote the independent variable by x instead of r. Also to avoid confusion, we denote by \(\epsilon\) the parameter multiplying the factor \(x^2\) in the potential instead of \(\lambda\) as in Ref. [7].

  8. We use Mathematica 12.0 in an AMD Ryzen 5 six core processor to 4200 MHz with 16 GB of RAM.

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Acknowledgements

This work was supported by CONAHCYT México, SNI México, EDI IPN, COFAA IPN, and Research Project IPN SIP 20240245. J. Jaimes-Najera appreciates the support of a scholarship awarded by CONAHCYT México.

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  1. J. Jaimes-Najera and A. Lopez-Ortega contributed equally to the paper.

Authors and Affiliations

  1. Departamento de Física, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Edificio 9, 07738, Ciudad de México, Ciudad de México, Mexico

    J. Jaimes-Najera & A. López-Ortega

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  1. J. Jaimes-Najera
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  2. A. López-Ortega
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Correspondence to A. López-Ortega.

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Jaimes-Najera, J., López-Ortega, A. Perturbation theory in the framework of the improved asymptotic iteration method. Eur. Phys. J. Plus 139, 259 (2024). https://doi.org/10.1140/epjp/s13360-024-04991-w

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