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The general relativistic hydrogen atom

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Abstract

The general relativistic Dirac equation is formulated in an arbitrary curved space-time using differential forms. These equations are applied to spherically symmetric systems with arbitrary charge and mass. For the case of a black hole (with event horizon) it is shown that the Dirac Hamiltonian is self-adjoint, has essential spectrum the whole real line and no bound states. Although rigorous results are obtained only for a spherically symmetric system, it is argued that, in the presence of any event horizon there will be no bound states. The case of a naked singularity is investigated with the results that the Dirac Hamiltonian is not self-adjoint. The self-adjoint extensions preserving angular momentum are studied and their spectrum is found to consist of an essential spectrum corresponding to that of a free electron plus eigenvalues in the gap (−mc 2, +mc 2). It is shown that, for certain boundary conditions, neutrino bound states exist.

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Authors and Affiliations

  1. Department of Physics, University of Pennsylvania, 19104, Philadelphia, PA, USA

    Jeffrey M. Cohen

  2. Mathematics Department, University of Pennsylvania, 19104, Philadelphia, PA, USA

    Robert T. Powers

Authors
  1. Jeffrey M. Cohen
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  2. Robert T. Powers
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Communicated by B. Simon

Supported in part by the National Science Foundation

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Cohen, J.M., Powers, R.T. The general relativistic hydrogen atom. Commun.Math. Phys. 86, 69–86 (1982). https://doi.org/10.1007/BF01205662

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  • DOI: https://doi.org/10.1007/BF01205662

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