## 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