Frank, Rupert L; Lieb, Élliott H; Seiringer, RobertIST Austria
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.
Communications in Mathematical Physics
405 - 424
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/s00220-012-1436-9
Frank, R., Lieb, É., & Seiringer, R. (2012). Binding of polarons and atoms at threshold. Communications in Mathematical Physics, 313(2), 405–424. https://doi.org/10.1007/s00220-012-1436-9
Frank, Rupert, Élliott Lieb, and Robert Seiringer. “Binding of Polarons and Atoms at Threshold.” Communications in Mathematical Physics 313, no. 2 (2012): 405–24. 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, pp. 405–424, 2012.
Frank R, Lieb É, Seiringer R. 2012. Binding of polarons and atoms at threshold. Communications in Mathematical Physics. 313(2), 405–424.
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.