  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. Fr\"ohlich.
The only parameters of this model, after removing the ultraviolet cutoff, are
the constants $U>0$ and $\alpha>0$ measuring the electron-electron and the
electron-phonon coupling strengths. They are constrained by the condition
$\sqrt{2}\alpha<U$, which follows from the dependence of $U$ and $\alpha$ on
electrical properties of the crystal. We show that the large $N$ asymptotic
behavior of the minimal energy $E_N$ changes at $\sqrt{2}\alpha=U$ and that
$\sqrt{2}\alpha\leq U$ is necessary for thermodynamic stability: for
$\sqrt{2}\alpha > U$ the phonon-mediated electron-electron attraction overcomes
the Coulomb repulsion and $E_N$ behaves like $-N^{7/3}$.
