Physics – Quantum Physics
Scientific paper
2011-05-20
Physics
Quantum Physics
16 pages, 1 figure
Scientific paper
The three-body Schr\"{o}dinger operator on the space of square integrable functions is found to be a certain extension of its unitary equivalent one whose exponential unitary group contains a subgroup with nilpotent Lie algebra of length $\kappa+1$, $\kappa\geq0$. As a result, the solutions to the three-body Schr\"{o}dinger equation are shown to exist in the commutator subalgebras. For the Coulomb three-body systems, it appears that the task is to solve---in these subalgebras---the radial Schr\"{o}dinger equation in three dimensions with the inverse power potential of the form of $r^{-\kappa-1}$. As an application to Coulombic systems, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge systems, obtained solutions, with $\kappa=0$, are reduced to the well-known eigenvalues of bound states at threshold.
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