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Level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators. (English) Zbl 1167.81382

Asch, Joachim (ed.) et al., Mathematical physics of quantum mechanics. Selected and refereed lectures from the QMath9 conference, Giens, France, September 12–16, 2004. Berlin: Springer (ISBN 3-540-31026-6/hbk). Lecture Notes in Physics 690, 383-402 (2006).
From the introduction: This report is devoted to the study of the spectral properties of the family of one-dimensional quasi-periodic Schrödinger operators acting on \(L ^{2}(\mathbb R)\) defined by
\[ H_{z,\varepsilon}\psi= -\frac{d^2}{dx^2} \psi(x)+ \big(V(x-z)+\alpha\cos(\varepsilon x)\big) \psi(x), \]
where
(H1) \(V:\mathbb R\to\mathbb R\) is a nonconstant, locally square integrable, 1-periodic function;
(H2) \(\varepsilon\) is a small positive number chosen such that \(2\pi/\varepsilon\) be irrational;
(H3) \(z\) is a real parameter indexing the operators;
(H4) \(\alpha\) is a strictly positive parameter.
For the entire collection see [Zbl 1130.81003].

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
70K43 Quasi-periodic motions and invariant tori for nonlinear problems in mechanics
70H11 Adiabatic invariants for problems in Hamiltonian and Lagrangian mechanics
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