Closed form representation for a projection onto infinitely-dimensional subspace spanned by Coulomb bound states

Abstract

The closed form integral representation for the projection onto the subspace spanned by bound states of the two-body Coulomb Hamiltonian is obtained. The projection operator onto the n2-dimensional subspace corresponding to the nth eigenvalue in the Coulomb discrete spectrum is also represented as the combination of Laguerre polynomials of nth and (n − 1)th order. The latter allows us to derive an analogue of the Christoffel–Darboux summation formula for the Laguerre polynomials. The representations obtained are believed to be helpful in solving the breakup problem in a system of three charged particles where the correct treatment of infinitely many bound states in two-body subsystems is one of the most difficult technical problems.