Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger operators (Q2043651)

The author proves several new, simple, and self-contained equivalences among various different descriptions of uniformly hyperbolic cocycles along with applications to Schrödinger cocycles and their relations to the spectral analysis of one-dimensional discrete Schrödinger operators. A general setting for this is that of a bijective map \(T\) on a set \(\Omega\), a map \(A:\Omega \to \mathrm{SL}(2;{\mathbb R})\) with \(\Vert A\Vert_\infty\) finite, and the dynamical system \[ (T,A):\Omega\times {\mathbb R}^2 \to \Omega\times {\mathbb R}^2, \ (T,A)(\omega,\vec{v}) = (T(\omega),A(\omega)\vec{v}). \] The map \((T,A)\) is iterated to give \((T,A)^n = (T^n,A_n)\) for \(n\in {\mathbb Z}\), where \(A_0(\omega) = I_2\), \(A_n(\omega) = A(T^{n-1}\omega)\cdots A(\omega)\) for \(n\geq 1\), and \(A_n(\omega) = [A_{-n}(T^n\omega)]^{-1}\) for \(n\leq -1\). The map \(A\) is a cocycle. The dynamical system \((T,A)\) is uniformly hyperbolic if there exist two maps \(u,s:\Omega \to \mathbb{RP}^1\) such that: (1) \(u,s\) are \((T,A)\)-invariant, i.e., for all \(\omega\in \Omega\) there holds \(A(\omega)\cdot u(\omega) = u[T\omega]\) and \(A(\omega)\cdot s(\omega) = s[T(\omega)]\); (2) There exist \(C>1\) and \(\lambda>1\) such that \(\Vert A_{-n}(\omega)\vec{v}\Vert \leq C\lambda^{-n}\) and \(\Vert A_n(\omega)\vec{w}\Vert \leq C\lambda^{-n}\) for all \(n\geq 1\), for all \(\omega\in \Omega\), and for all unit vectors \(\vec{v} \in u(\omega)\) and \(\vec{w}\in s(\omega)\). A special case of the general dynamical system \((T,A)\) and the cocycle \(A\) occurs when \(\Omega = {\mathbb Z}\) and \(T(j) = j+1\). Here the cocycle is the map \(A:{\mathbb Z}\to \mathrm{SL}(2,{\mathbb R})\) where the iteration is given by \(A_0 = I_2\), \(A_n(j) = A(j+n-1)\cdots A(j)\) for \(n\geq 1\), and \(A_{-n}(j) = A(j+n)^{-1}\cdots A(j-1)^{-1}\) for \(n\leq -1\). Cocycles of the form \(A:{\mathbb Z}\to \mathrm{SL}(2,{\mathbb R})\) are connected with one-dimensional Schrödinger operators. For a bounded potential \(v:{\mathbb Z}\to {\mathbb R}\), a one-dimensional Schrödinger operator \(H_v:\ell_2({\mathbb Z}) \to \ell_2({\mathbb Z})\) is given by \[ H_v(\psi)_n = \psi_{n+1} + \psi_{n-1} + v(n) \psi_n, \] for \(\psi = (\psi_n) \in \ell_2({\mathbb Z})\). With \(\Vert v\Vert_\infty < M\), the spectrum of the self-adjoint operator \(H_v\) is contained in the real interval \([-M-2,M+2]\). For \(E\) being the energy parameter, a \(\psi\in \ell_2({\mathbb Z})\) solves the spectral equation \(H_v \psi = E\psi\) if and only if the components \((\psi_n)\) of \(\psi\) satisfy \[ A^{(E-v)}(j)\begin{pmatrix} \psi_j \\ \psi_{j-1}\end{pmatrix} = \begin{pmatrix} \psi_{j+1} \\ \psi_j\end{pmatrix} \mathrm{ for \ all\ }j\in {\mathbb Z}, \] where \(A^{(E-v)}:{\mathbb Z}\to \mathrm{SL}(2,{\mathbb R})\) is the Schrödinger cocycle map defined by \[ A^{(E-v)}(j) = \begin{pmatrix} E-v(j) & -1 \\ 1 & 0\end{pmatrix}. \] There are five main results presented in the paper. (1) The first main theorem is for a cocycle \(A:{\mathbb Z}\to \mathrm{SL}(2,{\mathbb R})\). It states that \(A\) is uniformly hyperbolic if and only if there exists \(c>0\) and \(\lambda>0\) such that \(A\) satisfies the exponential growth condition, \(\Vert A_n(j)\Vert \geq c\lambda^n\) for all \(n\in {\mathbb Z}_+\) and for all \(j\in{\mathbb Z}\). (2) The second main theorem is for the dynamical system \((T,A)\), where it is assumed that \(\Omega\) is a compact metric space, \(T\) is a homeomorphism, and \(A:\Omega\to \mathrm{SL}(2,{\mathbb R})\) is continuous. It states that \((T,A)\) is not uniformly hyperbolic if and only if there is \(\omega\in \Omega\) and a unit vector \(\vec{v}\in {\mathbb R}^2\) such that \(\Vert A_n(\omega)\vec{v}\Vert\leq 1\) for all \(n\in{\mathbb Z}\). The latter condition is essentially the Sacker-Sell orbit. (3) The third main result, which is known as Johnson's Theorem, relates the dynamics of the Schrödinger cocycle and the spectral theory of Schrödinger operator. This result states that the spectrum of \(H_v\) is precisely those real values of \(E\) for which \(A^{(E-v)}\) is not uniformly hyperbolic. (4) The fourth main result requires embedding the operator \(H_v\) into a dynamically defined family of operators \((H_\omega)_{\omega\in \Omega}\). To do this one may start with an ergodic system \((\Omega,T,\mu)\) for a compact metric space \(\Omega\), a homeomorphism \(T\) of \(\Omega\), and \(T\)-ergodic probability measure on \(\Omega\). Then for a continuous function \(f:\Omega\to {\mathbb R}\) we define \(H_\omega\) by \[ (H_\omega \psi)_n = \psi_{n+1} + \psi_{n-1} + f(T^n\omega) \psi_n, \ \omega\in \Omega. \] The Schrödinger cocycle map is now \[ A^{(E-f)}(\omega) = \begin{pmatrix} E - f(\omega) & -1 \\ 1 & 0\end{pmatrix}, \ E\in {\mathbb R}. \] The fourth main theorem is the standard version of Johnson's Theorem. It assumes that \((\Omega,T)\) is transitive and chooses \(\omega_0\in \Omega\) such that \(\overline{\mathrm{Orb}(\omega_0)} = \Omega\). Then if \(\Sigma = \sigma(H_{\omega_0})\) (the spectrum of \(H_{\omega_0}\)), then for all \(\omega\in \Omega\) it holds that \(\sigma(H_\omega)\subset \Sigma\), and moreover that \[ \Sigma = \{ E : (T, A^{(E-f)}) \mathrm{\ is\ not\ uniformly\ hyperbolic}\}. \] (5) The fifth main result involves the avalanche principle and is valid for cocycles of the form \(A:{\mathbb Z}\to \mathrm{SL}(2,{\mathbb R})\). This result states that if there is \(\lambda>C\) , largeness independent of \(n\) below, so that for each \(j\in {\mathbb Z}\) there holds \(\Vert A(j)\Vert \geq \lambda\) and \[ \vert \log\Vert A(j+1)\Vert + \log\Vert A(j)\Vert - \log\Vert A(j+1)A(j)\Vert\,\vert \leq \frac{\log \lambda}{2}, \] then \(A\) is uniformly hyperbolic and for each \(j\in {\mathbb Z}\) and each \(n\in {\mathbb Z}_+\) there holds \[ \left\vert \log\Vert A_n(j)\Vert+\sum_{k=1}^{n-2} \log\Vert A(j+k)\Vert - \sum_{k=0}^{n-2} \Vert A(j+k+1)A(j+k)\Vert\,\right\vert\leq C \frac{n}{\lambda}. \]