Binding of Polarons and Atoms at Threshold

TL;DR

This paper proves that at the critical coupling constant, the size of polarons, bipolarons, and atoms reaches a maximum before exploding, using energy estimates and general principles rather than detailed Schrödinger analysis.

Contribution

It establishes a general result that the system size peaks at the critical coupling and then diverges, applicable to polarons, bipolarons, and atomic systems, using energy-based methods.

Findings

System size reaches a maximum at the critical coupling constant.

The phenomenon occurs for both polarons and atoms like helium.

Proofs rely on energy estimates, not detailed Schrödinger analysis.

Abstract

If the polaron coupling constant $\alpha$ is large enough, bipolarons or multi-polarons will form. When passing through the critical $\alpha_c$ from above, does the radius of the system simply get arbitrarily large or does it reach a maximum and then explodes? 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 $\alpha_c$. 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 Schr\"odinger 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 $1/r^2$.