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Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies
Puig Sadurní, Joaquim; Simó Torres, Carlos
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
n this article we investigate numerically the spectrum of some representative examples of discrete one-dimensional Schrödinger operators with quasi-periodic potential in terms of a perturbative constant b and the spectral parameter a. Our examples include the well-known Almost Mathieu model, other trigonometric potentials with a single quasi-periodic frequency and generalisations with two and three frequencies. We computed numerically the rotation number and the Lyapunov exponent to detect open and collapsed gaps, resonance tongues and the measure of the spectrum. We found that the case with one frequency was significantly different from the case of several frequencies because the latter has all gaps collapsed for a sufficiently large value of the perturbative constant and thus the spectrum is a single spectral band with positive Lyapunov exponent. In contrast, in the cases with one frequency considered, gaps are always dense in the spectrum, although some gaps may collapse either for a single value of the perturbative constant or for a range of values. In all cases we found that there is a curve in the (a, b)-plane which separates the regions where the Lyapunov exponent is zero in the spectrum and where it is positive. Along this curve, which is b = 2 in the Almost Mathieu case, the measure of the spectrum is zero.
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística
Schrödinger equation
Lyapunov exponents
Equacions diferencials parcials
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37B Topological dynamics
Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type
Attribution-NonCommercial-NoDerivs 3.0 Spain

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