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The spectral theory of adiabatic quasi-periodic operators on the real line. (English) Zbl 1059.34060

This survey paper summarizes spectral properties of the family \[ H_{z,\varepsilon} \psi:=-\frac{d^2}{dx^2}\psi(x)+[(V(x-z)+W(\varepsilon x)] \psi(x)=E\psi (x),\quad x\in \mathbb{R}, \] of differential equations with \(W(z)=\alpha \cos z\), \(\alpha >0\), where \(V\) is a nonconstant \(1\)-periodic function, \(\varepsilon >0\) is fixed, \(z\) is a real parameter indexing the family. It is well-known that the spectrum of the Schrödinger operator \(H_0=-\frac{d^2}{dx^2}+V\) consists of intervals \([E_1,E_2]\), \([E_3,E_4]\), …, with \(E_1<E_2\leq E_3<E_4\dots\). Throughout this paper, all spectral gaps are assumed to be proper, i. e., \(E_{2n}<E_{2n+1}\) for \(n=1,2,\dots\). Let \([W_-,W_+]=[-\alpha ,\alpha ]\) be the range of \(W\). Depending on the location of \(\mathcal W(E)=[E-W_+,E-W_-]\) with respect to the bands of the spectrum of \(H_0\), four distinct cases are considered. Properties of the spectrum in each case are discussed, with particular interest in the case of small positive \(\varepsilon \).

MSC:

34L05 General spectral theory of ordinary differential operators
34E05 Asymptotic expansions of solutions to ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
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