Binding of Polarons and Atoms at Threshold

Rupert L. Frank, Elliott H. Lieb, Robert Seiringer

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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 languageEnglish (US)
Pages (from-to)405-424
Number of pages20
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Jul 2012

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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