Peierls-Fröhlich problem and potentials with finite number of gaps. I

Exact solution of the Peierls–Fröhlich problem about the self-consistent state of conduction electron and lattice is proved to be a one-gap potential. Equations which describe the dependence of the boundaries of the spectrum on the parameters of the problem (such as the electron density, lattice elastic constant and temperature) are obtained. The equations are exactly solved at the absolute zero of temperature and investigated at the critical temperature at which lattice deformations arise. Charge density waves and condensons are shown to be limiting cases of the considered selfconsistent state.