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 \(\varepsilon \).


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