Weakly Resonant Tunneling Interactions for Adiabatic Quasi-periodic Schrödinger Operators

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. Résumé. 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 entrâıne 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 type spectraux i.e. une alternance entre du spectre singulier et du spectre absolument continu. Un autre phénomène observé est l’apparition d’̂ılots de spectre absolument continu dans du spectre singulier dûs aux résonances. 0. Introduction The present paper is devoted to the analysis of the family of one-dimensional quasi-periodic Schrödinger operators acting on L2(R) defined by (0.1) Hz,ε = − d2 dx2 + V (x− z) + α cos(εx). We assume that (H1): V : R → R is a non constant, locally square integrable, 1-periodic function; (H2): ε is a small positive number chosen such that 2π/ε be irrational; (H3): z is a real parameter; (H4): α is a strictly positive parameter that we will keep fixed in most of the paper. As ε is small, the operator (0.1) is a slow perturbation of the periodic Schrödinger operator (0.2) H0 = − d2 dx2 + V (x) acting on L2(R). To study (0.1), we use the asymptotic method for slow perturbations of onedimensional periodic equations developed in [11] and [9]. The results of the present paper are follow-ups on those obtained in [12, 8, 10] for the family (0.1). In these papers, we have seen that the spectral properties of Hz,ε at energy E depend crucially on the position of the spectral window F(E) := [E − α,E + α] with respect to the spectrum of the unperturbed operator H0. Note that the size of the window is equal to the amplitude of the adiabatic perturbation. In the present paper, the relative position is described in figure 1 i.e., we assume that there exists J , an interval of energies, such that, for all E ∈ J , the spectral window F(E) covers the edges of two neighboring spectral bands of H0 (see assumption (TIBM)). In this case, one can say that the spectrum in J is determined by the interaction of the neighboring spectral bands induced by the adiabatic perturbation. 1991 Mathematics Subject Classification. 34E05, 34E20, 34L05, 34L40.