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Títol: Resonance tongues and spectral gaps in quasi-periodic schrödinger operators with one or more frequencies. A numerical exploration
Autor: Puig Sadurní, Joaquim Veure Producció científica UPC; Simó Torres, Carlos Veure Producció científica UPC
Data: jul-2010
Tipus de document: External research report
Citació: [prepr201007PuiS]
Resum: Abstract. In this paper we investigate numerically the spectrum of some representative examples of discrete one-dimensional Schr¨odinger 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.
URI: http://hdl.handle.net/2117/8946
Versió de l'editor: http://www.ma1.upc.edu/~jpuig/preprints/puig-simo_10_1.pdf
Apareix a les col·leccions:EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions. Reports de recerca
Departaments de Matemàtica Aplicada. Reports de recerca
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