On the Bound States for the Three-Body Schrödinger Equation with Decaying Potentials

The three-body Schrödinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length κ+1, κ=0,1,... As a result, the solutions to the three-body Schrödinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve—in these subalgebras—the radial Schrödinger equation in three dimensions with the inverse power potential of the form r−κ−1. As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with κ=0, are reduced to the well-known eigenvalues of bound states at threshold. (author)