# Binding of polarons and atoms at threshold

Frank R, Lieb É, Seiringer R. 2012. Binding of polarons and atoms at threshold. Communications in Mathematical Physics. 313(2), 405–424.

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Author

Frank, Rupert L;
Lieb, Élliott H;
Seiringer, Robert

^{ISTA}^{}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.

Publishing Year

Date Published

2012-07-01

Journal Title

Communications in Mathematical Physics

Volume

313

Issue

2

Page

405 - 424

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### Cite this

Frank R, Lieb É, Seiringer R. Binding of polarons and atoms at threshold.

*Communications in Mathematical Physics*. 2012;313(2):405-424. doi:10.1007/s00220-012-1436-9Frank, R., Lieb, É., & Seiringer, R. (2012). Binding of polarons and atoms at threshold.

*Communications in Mathematical Physics*. Springer. https://doi.org/10.1007/s00220-012-1436-9Frank, Rupert, Élliott Lieb, and Robert Seiringer. “Binding of Polarons and Atoms at Threshold.”

*Communications in Mathematical Physics*. Springer, 2012. https://doi.org/10.1007/s00220-012-1436-9.R. Frank, É. Lieb, and R. Seiringer, “Binding of polarons and atoms at threshold,”

*Communications in Mathematical Physics*, vol. 313, no. 2. Springer, pp. 405–424, 2012.Frank, Rupert, et al. “Binding of Polarons and Atoms at Threshold.”

*Communications in Mathematical Physics*, vol. 313, no. 2, Springer, 2012, pp. 405–24, doi:10.1007/s00220-012-1436-9.**All files available under the following license(s):**

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