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Bounds on the Minimal Energy of Translation Invariant $N$-Polaron Systems

For systems of $N$ charged fermions (e.g. electrons) interacting with longitudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants $U>0$ and $α>0$ measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition $\sqrt{2}α<U$, which follows from the dependence of $U$ and $α$ on electrical properties of the crystal. We show that the large $N$ asymptotic behavior of the minimal energy $E_N$ changes at $\sqrt{2}α=U$ and that $\sqrt{2}α\leq U$ is necessary for thermodynamic stability: for $\sqrt{2}α> U$ the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and $E_N$ behaves like $-N^{7/3}$.