Equidistribution of Zeros of Dirichlet Determinants and Fine Properties of Integrated Density of States

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 function V (x). We study the eigenvalues of the problem H(x,ω)φ = Eφ on a finite interval [1, N ] with Dirichlet boundary conditions. We assume that the Lyapunov exponents of H(x,ω)φ(n) = Eφ(n), n ∈ Z are positive for all E. If V (x) is a trigonometric polynomial, then we show that for Diophantine ω, the averaged number of these Dirichlet eigenvalues which fall into an interval (E − η,E + η), η ≍ exp ( −(logN) ) , 0 < δ ≪ 1 does not exceed N exp ( (log η) ) η 1 2k0 , 0 < β ≪ 1, where k0 = deg V . Moreover, for typical ω and any E ∈ R \ E(η), mes E(η) < exp ( −(log η)1 ) , this averaged number does not exceed N exp ( (log η) ) η. For the integrated density of states N (·) of the problem H(x,ω)φ = Eφ, this implies that N (E + η) −N (E − η) ≤ exp ( (log η) ) η 1 2k0 for any E ∈ R, η > 0 and N (E + η)− N (E − η) ≤ exp ( (log ε) ) exp ( (log η) ) η for E ∈ R \ E , mes E (ε) < ε. To investigate the distribution of the Dirichlet eigenvalues of H(x,ω)φ = Eφ on a finite interval [1, N ] we study the distribution of the zeros of the characteristic determinants fN (·, ω,E) with complexified phase x, and frozen ω,E. We prove equidistribution of these zeros in some annulus Aρ = {z ∈ C : 1− ρ < |z| < 1 + ρ} and show also that no more than 2k0 of them fall into a disk of radius exp(−N), δ ≪ 1.