https://doi.org/10.1140/epja/i2007-10613-1
Regular Article - Theoretical Physics
Completeness of the Coulomb scattering wave functions
1
Cyclotron Institute, Texas A&M University, 77843, College Station, TX, USA
2
University of Georgia, Athens, GA, USA
Received:
28
March
2007
Revised:
24
April
2008
Published online:
25
July
2008
The completeness of the eigenfunctions of a self-adjoint Hamiltonian, which is the basic ingredient of quantum mechanics, plays an important role in nuclear reaction and nuclear-structure theory. Here we present the first formal proof of the completeness of the two-body Coulomb scattering wave functions for a repulsive unscreened Coulomb potential using Newton’s method (R. Newton, J. Math. Phys. 1, 319 (1960)). The proof allows us to claim that the eigenfunctions of the two-body Hamiltonian, with the potential given by the sum of the repulsive Coulomb plus short-range (nuclear) potentials, form a complete set. It also allows one to extend Berggren’s approach for the modification of the complete set of eigenfunctions by including the resonances for charged particles. We also demonstrate that the resonant Gamow functions with Coulomb tail can be regularized using Zel’dovich’s regularization method.
PACS: 03.65.Nk Scattering theory – / 03.65.Ca Formalism – / 24.30.-v Resonance reactions – / 21.60.Cs Shell model –
Communicated by U.-G. Meiβner
For the continuum spectrum the eigenfunctions are not square-integrable, strictly speaking we need to use a rigged Hilbert space which extends the normal Hilbert space by bringing together the discrete and continuum spectrum eigenstates.
© SIF, Springer-Verlag Berlin Heidelberg 2008
