Localization for a class of one dimensional quasi-periodic Schrödinger operators. (English) Zbl 0722.34070

The authors study the two operators \[ H=-\epsilon^ 2\Delta +(1/2\pi)\cos 2\pi (j\alpha +\theta)\text{ on } \ell^ 2({\mathbb{Z}}), \]
\[ H_ c=-d^ 2/dx^ 2-K^ 2(\cos 2\pi x+\cos 2\pi (\alpha x+\theta))\text{ on } L^ 2({\mathbb{R}}), \] where \(\alpha\) and \(\theta\) are fixed parameters, \(\epsilon\) is small enough, and K is large enough. Under some diophantine condition on \(\alpha\), they prove that H and \(H_ c\) have only pure point spectrum of almost all \(\theta\) in [0,2\(\pi\) ]. The main idea consists in constructing for any integer n and any energy level E a family \(S_ n\) of disjoint intervals, such that any generalized eigenfunction of energy E can be expressed (near infinity) in terms of the Green function of the Dirichlet realization of the operator, on some interval \(\Lambda\) disjoint from \(S_ n\). Moreover, this family \(S_ n\) is such that the Green function associated this way to any such interval \(\Lambda\), has exponentially small bounds.


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35P05 General topics in linear spectral theory for PDEs
34B27 Green’s functions for ordinary differential equations
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