A mathematical basis is given to the Peierls-Fr6hlich instability and the Kohn anomaly. The techniques and ideas are based on the recently developed mathematical theory of quantum fluctuations and response theory. We prove that there exists a unique resonant one-mode interaction between electrons and phonons which is responsible for the Peierls-Fr/Shlich instability and the phase transition in the Mattis-Langer model. We prove also that the softening of this phonon mode at the critical temperature (Kohn anomaly) is a consequence of the critical slowing down of the dynamics of the lattice distortion fluctuations. It is the result of the linear dependence of two fluctuation operators corresponding to the frozen charge density wave and the distortion order parameter.