Binding of Polarons and Atoms at Threshold

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 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 $1/r^2$.

19 pages

References in corpus (4)

• Geometric methods for nonlinear many-body quantum systems
• Bi-Polaron and N-Polaron Binding Energies
• The bipolaron in the strong coupling limit
• Bounds on the Minimal Energy of Translation Invariant $N$-Polaron Systems

Cited by in corpus (6)

• Derivation of Pekar's Polarons from a Microscopic Model of Quantum Crystals
• Bound States at Threshold resulting from Coulomb Repulsion
• Existence of ground states for negative ions at the binding threshold
• Monotonicity of the polaron energy II:General theory of operator monotonicity
• Symmetry of bipolaron bound states for small Coulomb repulsion
• Ground State of the Polaron Hydrogenic Atom in a Strong Magnetic Field