PEIERLS-FROHLICH INSTABILITY AND KOHN ANOMALY

A mathematical basis is given to the Peierls-Frohlich instability and the Kohn anomaly. The techniques and ideas are based on the recently d eveloped mathematical theory of quantum fluctuations and response theo ry. We prove that there exists a unique resonant one-mode interaction between electrons and phonons which is responsible for the Peierls Fro hlich 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 do wn of the dynamics of the lattice distortion fluctuations. It is the r esult of the linear dependence of two fluctuation operators correspond ing to the frozen charge density wave and the distortion order paramet er.