Quantum particle in a one-dimensional deformed lattice. Estimates of the gaps in the spectrum

It is shown that the potential of oscillating lattice at fixed moment of time is a quasi-periodical function. Stationary states of quantum particle in quasi-periodical potential satisfy the generalized Floquet–Bloch theorem and can be characterized by the quasi-momentum which is connected with the density of states exactly in the same way as in the case of the particle in a purely periodical potential. If the quasi-momentum satisfies the generalized Bragg–Woolfe conditions, the spectrum may include the lacunae (forbidden zones), for the sizes of which the estimates are given.