Abstract
The Schr\"odinger equation for the tight-binding model is solved in the continuum limit. It is found that the modulated electron wave function takes the form of the Jacobian elliptic function with periodicity varied continuously as a function of the electron-phonon coupling \ensuremath{\alpha}. This property may account for the incommensurate modulation of the charge-density wave in the lattice, a fact that leads to the Fr\"ohlich superconductivity. For \ensuremath{\alpha}>${\ensuremath{\alpha}}_{c}$, the transition caused by the breaking of analyticity observed numerically by Aubry is shown to emerge from the poles of the charge density.
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