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One-dimensional
hydrogenic ions with screened nuclear Coulomb field
Suchindram Dasgupta, Chirag Khurana and A. Shadi Tahvildar-Zadeh |
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Vol. 19 (2026), No. 4, 593–632
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Abstract |
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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 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. |
Keywords
Dirac's equation, relativistic Hamiltonian, hydrogen
spectrum, one-dimensional quantum system
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Mathematical Subject Classification
Primary: 34B24, 35Q40, 37J46, 47A75, 81Q10
Secondary: 37L10
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Milestones
Received: 30 July 2024
Revised: 2 March 2025
Accepted: 10 March 2025
Published: 13 October 2026
Communicated by Martin Bohner
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Authors |
| Suchindram Dasgupta | |
| Department of Physics and
Astronomy West Virginia University Morgantown, WV United States |
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| Chirag Khurana | |
| Applied Physics & Applied
Mathematics Department Columbia University New York, NY United States |
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| A. Shadi Tahvildar-Zadeh | |
| Department of Mathematics Rutgers University (New Brunswick) Piscataway, NJ United States |
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| © 2026 MSP (Mathematical Sciences Publishers). |
