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Adiabatic almost-periodic Schrödinger operators. (English. Russian original) Zbl 1237.34147

J. Math. Sci., New York 173, No. 3, 299-319 (2011); translation from Zap. Nauchn. Semin. POMI 379, 103-141 (2010).
The author considers the ergodic operator family \[ H=-\frac{d^{2}}{dx^{2}}+V(x-z)+W(\varepsilon x),\quad x\in \mathbb{R}, \] where \(V:\mathbb{R}\times\mathbb{R}\) is a nonconstant, locally square integrable, \(1\)-periodic function, \( 0<\varepsilon <2\pi \) is the frequency, a constant such that the ratio \(\frac{ 2\pi }{\varepsilon }\) is irrational; \(0\leq z<2\pi \) is the ergodic parameter; \(W\) is a nonconstant \(2\pi\)-periodic function analytic in a neighborhood of the real line and taking real values along the real line. Here, the adiabatic case is discussed when the number \(\varepsilon\) is small and one of the frequencies is much greater than the other. In the study, the monodromy matrix and the finite difference monodromy equations are used. Spectral results are presented by using the local asymptotics of the solutions of the almost periodic equation, described by the asymptotic functional structure of the monodromy equation, a finite difference equation determined by the monodromy matrix.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators
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