In this paper, the Schrödinger equation in the presence of a screened Coulomb potential and a uniform field is analysed using matched asymptotic expansions. When the cup well potential has a very short range, approximate analytical expressions for the energy levels and the lifetime of the system are found. The results are compared with those described in the literature. This paper may be helpful for undergraduate and graduate students in physics as an introductory problem in the application of asymptotic matching applied to quantum mechanics.
The circular Sitnikov problem is revisited, using matched asymptotic expansions. In the case of large oscillation periods, approximate analytical expressions for the period and the orbit of the third body are found. The results are compared with those described in the literature and show that the movement of the third body can be well described by two analytical solutions, the inner and outer solutions.
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