### Abstract

If the polaron coupling constant α is large enough, bipolarons or multi-polarons will form. When passing through the critical α _{c} from above, does the radius of the system simply get arbitrarily large or does it reach a maximum and then explode? We prove that it is always the latter. We also prove the analogous statement for the Pekar-Tomasevich (PT) approximation to the energy, in which case there is a solution to the PT equation at α _{c}. Similarly, we show that the same phenomenon occurs for atoms, e. g., helium, at the critical value of the nuclear charge. Our proofs rely only on energy estimates, not on a detailed analysis of the Schrödinger equation, and are very general. They use the fact that the Coulomb repulsion decays like 1/r, while 'uncertainty principle' localization energies decay more rapidly, as 1/r ^{2}.

Original language | English (US) |
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Pages (from-to) | 405-424 |

Number of pages | 20 |

Journal | Communications In Mathematical Physics |

Volume | 313 |

Issue number | 2 |

DOIs | |

State | Published - Jul 2012 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

*313*(2), 405-424. https://doi.org/10.1007/s00220-012-1436-9