Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrödinger operators
Mémoires de la Société Mathématique de France, no. 104 (2006) , 119 p.

In this paper, we study spectral properties of the one dimensional periodic Schrödinger operator with an adiabatic quasi-periodic perturbation. We show that in certain energy regions the perturbation leads to resonance effects related to the ones observed in the problem of two resonating quantum wells. These effects affect both the geometry and the nature of the spectrum. In particular, they can lead to the intertwining of sequences of intervals containing absolutely continuous spectrum and intervals containing singular spectrum. Moreover, in regions where all of the spectrum is expected to be singular, these effects typically give rise to exponentially small “islands” of absolutely continuous spectrum.

Cet article est consacré à l’étude du spectre d’une famille d’opérateurs quasi-périodiques obtenus comme perturbations adiabatiques d’un opérateur périodique fixé. Nous montrons que, dans certaines régions d’énergies, la perturbation entraîne des phénomènes de résonance similaires à ceux observés dans le cas de deux puits. Ces effets s’observent autant sur la géométrie du spectre que sur sa nature. En particulier, on peut observer un entrelacement de types spectraux i.e. une alternance entre du spectre singulier et du spectre absolument continu. Un autre phénomène observé est l’apparition d’îlots de spectre absolument continu dans du spectre singulier dus aux résonances.

DOI: 10.24033/msmf.416
Classification: 34E05, 34E20, 34L05, 34L40
Keywords: Quasi-periodic Schrödinger equation, two resonating wells, pure point spectrum, absolutely continuous spectrum, complex WKB method, monodromy matrix
Mot clés : Équations de Schrödinger quasi-périodique, double puis résonnant, spectre purement ponctuel, spectre absolument continu, méthode BKW complexe, matrice de monodromie
     author = {Fedotov, Alexander and Klopp, Fr\'ed\'eric},
     title = {Weakly resonant tunneling interactions for adiabatic quasi-periodic {Schr\"odinger} operators},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {104},
     year = {2006},
     doi = {10.24033/msmf.416},
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     url = {http://www.numdam.org/item/MSMF_2006_2_104__1_0/}
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Fedotov, Alexander; Klopp, Frédéric. Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrödinger operators. Mémoires de la Société Mathématique de France, Serie 2, no. 104 (2006), 119 p. doi : 10.24033/msmf.416. http://numdam.org/item/MSMF_2006_2_104__1_0/

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