Geodesic
Quantum particle in a~one-dimensional deformed lattice. Estimates of the gaps in the spectrum
Teoretičeskaâ i matematičeskaâ fizika, Tome 25 (1975) no. 3, pp. 344-357

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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. 
@article{TMF_1975_25_3_a5,
     author = {E. D. Belokolos},
     title = {Quantum particle in a~one-dimensional deformed lattice. {Estimates} of the gaps in the spectrum},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {344--357},
     publisher = {mathdoc},
     volume = {25},
     number = {3},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1975_25_3_a5/}
}
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E. D. Belokolos. Quantum particle in a~one-dimensional deformed lattice. Estimates of the gaps in the spectrum. Teoretičeskaâ i matematičeskaâ fizika, Tome 25 (1975) no. 3, pp. 344-357. http://geodesic.mathdoc.fr/item/TMF_1975_25_3_a5/