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

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

2006-08-24

arxiv.org/abs/physics/0608245v1

, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 4767-4773

Physics

Atomic Physics

7 pages

Scientific paper

10.1088/0953-4075/39/22/019

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 $n^2$ dimensional subspace corresponding to the $n$-th eigenvalue in the Coulomb discrete spectrum is also represented as the combination of Laguerre polynomials of $n$-th and $(n-1)$-th order. The latter allows us to derive an analog 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.

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