Fine Properties of the Integrated Density of States and a Quantitative Separation Property of the Dirichlet Eigenvalues

Abstract.

We consider one-dimensional difference Schrödinger equations $[H(x,\omega)\varphi](n)\equiv - \varphi(n-1)-\varphi(n+1)+V(x+n\omega)\varphi(n) = E \varphi (n), n \, \in \, \mathbb {Z}, \, x, \omega \, \in \, [0,1]$ with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V ₀(x) of degree k ₀, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states $\mathcal{N}$ is Hölder $\frac{1}{2k_0}-k$ continuous for any k > 0. Moreover, we show that $\mathcal{N}$ is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem $H(x, \omega)\varphi = E \varphi$ on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval $(E - \eta, E + \eta) \, \rm{with}\, \eta \, \asymp \, N^{-1+\delta}, 0 < \delta \ll 1$, does not exceed $N\eta^{ \frac{1}{2k_0}-k}$, k > 0. Moreover, for $\omega\, \notin\, \Omega(\varepsilon), \rm {mes}\, \Omega(\varepsilon) \, < \,\varepsilon \, {\rm and}\,\, E \, \notin \, \mathcal{E}_{\omega} (\varepsilon), \, {\rm mes}\, \mathcal{E}_{\omega} (\varepsilon) \, < \, \varepsilon$, this averaged number does not exceed exp $((\log \varepsilon^{-1})^A)\eta N$, for any ${\eta} > N^{-1+b}, b > 0$. For the integrated density of states $\mathcal{N}(\cdot)$ of the problem $H(x, \omega) \varphi =E \varphi$ this implies that $\mathcal{N}(E + \eta)- \mathcal{N}(E - \eta) \leq {\rm exp}((\log \varepsilon^{-1})^A)\eta$ for any $E \, \notin \, {\mathcal{E}_\omega}(\varepsilon)$. To investigate the distribution of the Dirichlet eigenvalues of $H(x, \omega)\varphi = E \varphi$ on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants $f_N(\cdot,\omega, E)$ with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus $\mathcal{A}_\rho = \{z \, \in \, \mathbb{C} : 1-\rho < |z| < 1+\rho \}$ and show also that no more than 2k ₀ of them fall into any disk of radius exp$(-(\log N)^A), A {\gg}1$. In addition, we obtain the lower bound $e^{-{N}^{\delta}}$ (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies.