@article{2400, 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.}, author = {Frank, Rupert L and Lieb, Élliott H and Robert Seiringer}, journal = {Communications in Mathematical Physics}, number = {2}, pages = {405 -- 424}, publisher = {Springer}, title = {{Binding of polarons and atoms at threshold}}, doi = {10.1007/s00220-012-1436-9}, volume = {313}, year = {2012}, }