Abstract
We numerically investigate the following two-body stationary Schrödinger equation (SE): \(\left\{ { - \frac{{{{\rlap{--} h}^2}}}{{2m}}\frac{\partial }{{x_1^2}} - \frac{{{{\rlap{--} h}^2}}}{{2m}}\frac{\partial }{{x_2^2}} + V\left( {{x_2}} \right) + g\partial \left( {{x_1} - {x_2}} \right)} \right\}x\Psi \left( {{x_1},{x_2}} \right) = E\Psi \left( {{x_1},{x_2}} \right),\) where \({x_1},{x_2} \in \mathbb{R},V\left( {{x_1}} \right) = {V_0}{\sin ^2}\left( {{k_x}{x_i}} \right)\) is the potential describing the interaction of atoms with a trap and gδ(x 1–x 2) is the interatomic potential. Previously, a similar problem has been solved analytically for the harmonic interaction \({V_h}\left( {{x_i}} \right) = \frac{1}{2}m\omega x_i^2\) with the trap, which leads to the separation of coordinates \(y = \frac{{{x_1} + {x_2}}}{{\sqrt 2 }},{\kern 1pt} x = \frac{{{x_1} - {x_2}}}{{\sqrt 2 }}\) for the center-of-mass and relative motion. The anharmonicity of the trap couples these motions and, there-fore, the problem becomes significantly more complicated. In previous works, the anharmonicity V a = V–V h of the trap has been taken into account in the framework of perturbation theory. In this work, the energy level shifts of a two-body atomic system are calculated beyond the perturbation theory for different magnitudes of parameter g of the interatomic interaction. The results are compared to those computed using the perturbation theory.
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Original Russian Text © I.S. Ishmukhamedov, D.T. Aznabayev, S.A. Zhaugasheva, 2015, published in Pis’ma v Zhurnal Fizika Elementarnykh Chastits i Atomnogo Yadra, 2015.
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Ishmukhamedov, I.S., Aznabayev, D.T. & Zhaugasheva, S.A. Two-body atomic system in a one-dimensional anharmonic trap: The energy spectrum. Phys. Part. Nuclei Lett. 12, 680–688 (2015). https://doi.org/10.1134/S1547477115050076
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DOI: https://doi.org/10.1134/S1547477115050076