One-dimensional hydrogenic ions with screened nuclear Coulomb field
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916339998785536 |
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| author | Dasgupta, Suchindram Khurana, Chirag Tahvildar-Zadeh, A. Shadi |
| author_facet | Dasgupta, Suchindram Khurana, Chirag Tahvildar-Zadeh, A. Shadi |
| contents | We study the spectrum of the Dirac hamiltonian in one space dimension for a single electron in the electrostatic potential of a point nucleus, in the Born-Oppenheimer approximation where the nucleus is assumed fixed at the origin. The potential is screened at large distances so that it goes to zero exponentially at spatial infinity. We show that the hamiltonian is essentially self-adjoint, the essential spectrum has the usual gap $(-mc^2,mc^2)$ in it, and that there are only finitely many eigenvalues in that gap, corresponding to ground and excited states for the system. We find a one-to-one correspondence between the eigenfunctions of this hamiltonian and the heteroclinic saddle-saddle connectors of a certain dynamical system on a finite cylinder. We use this correspondence to study how the number of bound states changes with the nuclear charge. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_04033 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | One-dimensional hydrogenic ions with screened nuclear Coulomb field Dasgupta, Suchindram Khurana, Chirag Tahvildar-Zadeh, A. Shadi Mathematical Physics Analysis of PDEs Quantum Physics 35Q41, 81Q05 We study the spectrum of the Dirac hamiltonian in one space dimension for a single electron in the electrostatic potential of a point nucleus, in the Born-Oppenheimer approximation where the nucleus is assumed fixed at the origin. The potential is screened at large distances so that it goes to zero exponentially at spatial infinity. We show that the hamiltonian is essentially self-adjoint, the essential spectrum has the usual gap $(-mc^2,mc^2)$ in it, and that there are only finitely many eigenvalues in that gap, corresponding to ground and excited states for the system. We find a one-to-one correspondence between the eigenfunctions of this hamiltonian and the heteroclinic saddle-saddle connectors of a certain dynamical system on a finite cylinder. We use this correspondence to study how the number of bound states changes with the nuclear charge. |
| title | One-dimensional hydrogenic ions with screened nuclear Coulomb field |
| topic | Mathematical Physics Analysis of PDEs Quantum Physics 35Q41, 81Q05 |
| url | https://arxiv.org/abs/2312.04033 |