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Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique). (Semiclassical analysis for the Harper equation (with application to the magnetic Schrödinger equation)). (French) Zbl 0714.34130
[For part II see the paper reviewed in Zbl 0714.34131.] The authors study the Harper operator (H.0) cos(hD\({}_ x)+\cos x\) in \({\mathbb{R}}\) and the magnetic Schrödinger operator \[ (M.S.O)\quad (h_ 0D_ x-tA_ 1)^ 2+(h_ 0D_ x-tA_ 2)^ 2+V(x)\text{ on } {\mathbb{R}}^ 2. \] The main theorems in the paper, give upper and lower bounds on the spectrum of these operators. These bounds rely on the continued fraction expansion of h/2\(\pi\) (H.0 case) and \(2\pi\) ho/\(\phi\) t (M.S.O case) were \(\phi\) is the magnetic field through a basic periodic cell of V.
Using interaction matrices they show first that, except for a small forbidden zone, one can divide the spectrum of H.O into smaller sets where (modulo a renormalization) the H.O is unitary equivalent to a small perturbation of a new H.O. with a new constant \(h_ 1\) to which the analysis holds. The proof is then by induction.
For M.S.O. using Floquet theory when \(2\pi h_ 0/\phi t\) is rational, they study the band structure of the spectrum and the link with H.O. when \(2\pi h_ 0/\phi t\) is irrational they show that, restricted to a certain energy interval and modulo a renormalization, the M.S.O is unitary equivalent to a small perturbation of a H.O.
Reviewer: C.Zuily

MSC:
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35P15 Estimates of eigenvalues in context of PDEs
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35B20 Perturbations in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
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