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Continuous quasiperiodic Schrödinger operators with Gordon type potentials. (English) Zbl 1391.81083

Summary: Let us concern the quasi-periodic Schrödinger operator in the continuous case \((Hy)(x) = -y''(x) + V(x, \omega x)y(x)\), where \(V :(\mathbb{R} / \mathbb{Z})^2 \rightarrow \mathbb{R}\) is piecewisely \(\gamma\)-Hölder continuous with respect to the second variable. Let \(L(E)\) be the Lyapunov exponent of \(Hy = Ey\). Define \(\beta(\omega)\) as \(\beta(\omega) = \mathrm{lim sup}_{k \rightarrow \infty} \frac{- \ln \| k \omega \|}{k}\). We prove that \(H\) admits no eigenvalue in regime \(\{E \in \mathbb{R} | L(E) < \gamma \beta(\omega) \}\).{
©2018 American Institute of Physics}

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
26A16 Lipschitz (Hölder) classes
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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