Fedotov, Alexander; Klopp, Frédéric 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]. Cited in 2 Documents 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 PDFBibTeX XMLCite \textit{A. Fedotov} and \textit{F. Klopp}, Lect. Notes Phys. 690, 383--402 (2006; Zbl 1167.81382) Full Text: DOI arXiv