dorsal/arxiv
View SchemaCompleteness of the Coulomb scattering wave functions
| Authors | A. M. Mukhamedzhanov, M. Akin |
|---|---|
| Categories | |
| ArXiv ID | nucl-th/0602006 |
| URL | https://arxiv.org/abs/nucl-th/0602006 |
| DOI | 10.1140/epja/i2007-10613-1 |
| Journal | Eur.Phys.J.A37:185-192,2008 |
Abstract
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. However, until now, there was no a formal proof of the completeness of the eigenfunctions of the two-body Hamiltonian with the Coulomb interaction. Here we present the first formal proof of the completeness of the two-body Coulomb scattering wave functions for repulsive unscreened Coulomb potential. To prove the completeness we use the 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 also form a complete set. It also allows one to extend the Berggren's approach of modification of the complete set of the eigenfunctions by including the resonances for charged particles. We also demonstrate that the resonant Gamow functions with the Coulomb tail can be regularized using Zel'dovich's regularization method.
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"abstract": "Completeness of the eigenfunctions of a self-adjoint Hamiltonian, which is\nthe basic ingredient of quantum mechanics, plays an important role in nuclear\nreaction and nuclear structure theory. However, until now, there was no a\nformal proof of the completeness of the eigenfunctions of the two-body\nHamiltonian with the Coulomb interaction. Here we present the first formal\nproof of the completeness of the two-body Coulomb scattering wave functions for\nrepulsive unscreened Coulomb potential. To prove the completeness we use the\nNewton\u0027s method [R. Newton, J. Math Phys., 1, 319 (1960)]. The proof allows us\nto claim that the eigenfunctions of the two-body Hamiltonian with the potential\ngiven by the sum of the repulsive Coulomb plus short-range (nuclear) potentials\nalso form a complete set. It also allows one to extend the Berggren\u0027s approach\nof modification of the complete set of the eigenfunctions by including the\nresonances for charged particles. We also demonstrate that the resonant Gamow\nfunctions with the Coulomb tail can be regularized using Zel\u0027dovich\u0027s\nregularization method.",
"arxiv_id": "nucl-th/0602006",
"authors": [
"A. M. Mukhamedzhanov",
"M. Akin"
],
"categories": [
"nucl-th",
"math-ph",
"math.MP"
],
"doi": "10.1140/epja/i2007-10613-1",
"journal_ref": "Eur.Phys.J.A37:185-192,2008",
"title": "Completeness of the Coulomb scattering wave functions",
"url": "https://arxiv.org/abs/nucl-th/0602006"
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