## The Schrödinger equation with a quasi-periodic potential.(English)Zbl 0712.34094

The author considers the Schrödinger equation $-\frac{d^ 2}{dx^ 2}\psi +\epsilon (\cos x+\cos (\alpha x+\theta))\cdot \psi =E\psi$ where $$\epsilon$$ is small and $$\alpha$$ satisfies the diophantine inequality $$| p+q\alpha | \geq c/q^ 2$$ for p,q$$\in {\mathbb{Z}}$$, $$q\neq 0$$. He checks solutions of the form $\psi (x)=e^{ikx}\cdot q(x)=e^{iKx}\sum \psi_{mn}e^{inx}\cdot e^{im(\alpha x+\theta)}$ and tries to solve for $$\psi =\psi_{mn}$$ and he is led to the Schrödinger equation on the lattice $${\mathbb{Z}}^ 2$$ $H(K)\psi =(\epsilon \Delta +V(K))\psi =E\psi$ where $$\Delta$$ is the discrete Laplacian (without diagonal terms) and V(K) is some potential on $${\mathbb{Z}}^ 2$$. The main results are the following:
- for $$\epsilon$$ sufficiently small H(K) has pure point spectrum for almost every K,
- for $$\epsilon$$ sufficiently small the operator
$-d^ 2/dx^ 2+\epsilon (\cos x+\cos (\alpha x+\theta))$
has no point spectrum.
Reviewer: S.Balint

### MSC:

 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34D10 Perturbations of ordinary differential equations

### Keywords:

Schrödinger equation; point spectrum
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