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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
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