On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations | Annals of Mathematics

Abstract We consider one-dimensional difference Schrödinger equations [H(x,ω)φ](n)≡−φ(n−1)−φ(n+1)+V(x+nω)φ(n)=Eφ(n), n∈ℤ, x,ω∈[0,1] with real-analytic potential function V(x). If L(E,ω0) is greater than 0 for all E∈(E′,E”) and some Diophantine ω0, then the integrated density of states is absolutely continuous for almost every ω close to ω0, as shown by the authors in earlier work. In this paper we establish the formation of a dense set of gaps in spec(H(x,ω))∩(E′,E”). Our approach is based on an induction on scales argument, and is therefore both constructive as well as quantitative. Resonances between eigenfunctions of one scale lead to “pre-gaps” at a larger scale. To pass to actual gaps in the spectrum, we show that these pre-gaps cannot be filled more than a finite (and uniformly bounded) number of times. To accomplish this, one relates a pre-gap to pairs of complex zeros of the Dirichlet determinants off the unit circle. Amongst other things, we establish a nonperturbative version of the co-variant parametrization of the eigenvalues and eigenfunctions via the phases in the spirit of Sinai’s (perturbative) description of the spectrum via his function Λ. This allows us to relate the gaps in the spectrum with the graphs of the eigenvalues parametrized by the phase. Our infinite volume theorems hold for all Diophantine frequencies ω up to a set of Hausdorff dimension zero.

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数学年刊(Annals of Mathematics)

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