## 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.

### MSC:

 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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### References:

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