Three-Body Coulomb Problem

Abstract : We present a general approach for the solution of the three-body problem for a general interaction and apply it to the case of the Coulomb interaction. This approach is exact, simple, and fast. It makes use of integral equations derived from the consideration of the scattering properties of the system. In particular, this makes full use of the solution of the two-body problem, the interaction appearing only through the corresponding known T matrix. In the case of the Coulomb potential, we make use of a very convenient expression for the T matrix obtained by Schwinger. As a check, we apply this approach to the well-known problem of the helium atom ground state and obtain a perfect numerical agreement with the known result for the ground-state energy. The wave function is directly obtained from the corresponding solution. We expect our method to be, in particular, quite useful for the trion problem in semiconductors.
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Physical Review X, American Physical Society, 2017, 7 (4), pp.041035. 〈10.1103/PhysRevX.7.041035〉
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R. Combescot. Three-Body Coulomb Problem. Physical Review X, American Physical Society, 2017, 7 (4), pp.041035. 〈10.1103/PhysRevX.7.041035〉. 〈hal-01651622〉

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