### Abstract

We consider the quantum mechanical many-body problem of electrons and fixed nuclei interacting via Coulomb forces, but with a relativistic form for the kinetic energy, namely p^{2}/2 m is replaced by (p^{2}c^{2}+m^{2}c^{4})^{1/2}-mc^{2}. The electrons are allowed to have q spin states (q=2 in nature). For one electron and one nucleus instability occurs if zα>2/π, where z is the nuclear charge and α is the fine structure constant. We prove that stability occurs in the many-body case if zα≦2/π and α<1/(47 q). For small z, a better bound on α is also given. In the other direction we show that there is a critical α_{c} (no greater than 128/15π) such that if α>α_{c} then instability always occurs for all positive z (not necessarily integral) when the number of nuclei is large enough. Several other results of a technical nature are also given such as localization estimates and bounds for the relativistic kinetic energy.

Original language | English (US) |
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Pages (from-to) | 177-213 |

Number of pages | 37 |

Journal | Communications in Mathematical Physics |

Volume | 118 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1988 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications in Mathematical Physics*,

*118*(2), 177-213. https://doi.org/10.1007/BF01218577