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 TMF, 1975, Volume 25, Number 3, Pages 344–357 (Mi tmf4079)

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

E. D. Belokolos

Abstract: 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.

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English version:
Theoretical and Mathematical Physics, 1975, 25:3, 1176–1184

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Citation: E. D. Belokolos, “Quantum particle in a one-dimensional deformed lattice. Estimates of the gaps in the spectrum”, TMF, 25:3 (1975), 344–357; Theoret. and Math. Phys., 25:3 (1975), 1176–1184

Citation in format AMSBIB
\Bibitem{Bel75} \by E.~D.~Belokolos \paper Quantum particle in a~one-dimensional deformed lattice. Estimates of the gaps in the spectrum \jour TMF \yr 1975 \vol 25 \issue 3 \pages 344--357 \mathnet{http://mi.mathnet.ru/tmf4079} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=523387} \transl \jour Theoret. and Math. Phys. \yr 1975 \vol 25 \issue 3 \pages 1176--1184 \crossref{https://doi.org/10.1007/BF01040126} 

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This publication is cited in the following articles:
1. E. D. Belokolos, “Quantum particle in a one-dimensional deformed lattice. Dependence of the energy on the quasimomentum”, Theoret. and Math. Phys., 26:1 (1976), 21–25
2. M. A. Shubin, “Almost periodic functions and partial differential operators”, Russian Math. Surveys, 33:2 (1978), 1–52
3. M. A. Shubin, “The spectral theory and the index of elliptic operators with almost periodic coefficients”, Russian Math. Surveys, 34:2 (1979), 109–157
4. Ya. G. Sinai, “Structure of the spectrum of the Schrödinger operator with almost-periodic potential in the vicinity of its left edge”, Funct. Anal. Appl., 19:1 (1985), 34–39
5. E. I. Dinaburg, “Stark effect for a difference Schrödinger operator”, Theoret. and Math. Phys., 78:1 (1989), 50–57
6. E. I. Dinaburg, “Some questions of the spectral theory of discrete operators with quasiperiodic coefficients”, Russian Math. Surveys, 52:3 (1997), 451–499
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