On Lyapunov exponents of continuous Schrödinger cocycles over irrational rotations (Q2884415)

The authors consider the cocycle \(A(n, \theta)\) generated by a continuous \(\text{SL}(2, \mathbb{R})\)-valued map \(A\) defined on the unit circle \(\mathbb{T}\) by NEWLINE\[NEWLINE A(n, \theta)= \prod_{k=0}^{n-1} \, A^{\epsilon (n)} (\theta + k \alpha ),NEWLINE\]NEWLINE with \(\epsilon (n)= \operatorname{sgn} (n)\) and \(n \in \mathbb{Z}\), and where the frequency \(\alpha\) is a fixed irrational number.NEWLINENEWLINEThe cocycle admits a well-defined Lyapunov exponent given by NEWLINE\[NEWLINE \Lambda (A)= \lim_{n \rightarrow \infty} \frac{1}{n} \int_{\mathbb{T}} \, \log \|A(n, \theta) \| \, d\theta = \inf \int_{\mathbb{T}} \, \log \|A(n, \theta) \| \, d\theta. NEWLINE\]NEWLINE If \(\Lambda (A)>0\) then the corresponding co-cycle \(A(n, \theta)\) is said to be unitarily hyperbolic if \( \lim_{n\rightarrow \infty} \log \|A(n, \theta)\| = \Lambda (A)\) uniformly on \(\theta\). In this paper, the authors prove two theorems concerning the Schr\(\ddot{\text{o}}\)dinger cocycle with frequency \(\alpha \in \mathbb{R} \setminus \mathbb{Q}\) and generated by the family of \(\text{SL}(2, \mathbb{R})\)-valued continuous functions NEWLINE\[NEWLINE A_{f,E}(\theta)= \left( \begin{matrix} E-f(\theta) & -1 \\ 1 & 0 \end{matrix} \right)NEWLINE\]NEWLINE with \(E \in \mathbb{R}\) and \(f \in C(\mathbb{T})\). NEWLINENEWLINENEWLINE Theorem. There is a residual set of functions \(C(\mathbb{T})\), such that for any \(f \in C(\mathbb{T})\), one has NEWLINE\[NEWLINE \inf_{E \in \mathbb{R}} \Lambda (A_{\lambda f, E} )=0, \,\,\,\forall \, \lambda >0.NEWLINE\]NEWLINE Theorem. The set \(\{ f \in C(\mathbb{T}): \,\, A_{\lambda f} (n, \cdot)\,\, \text{is uniformly hyperbolic or} \,\, \Lambda_{\lambda f}=0 \,\,\,\text{for any}\,\, \lambda \in (0, \, \infty)\}\) is residual.NEWLINENEWLINEThese two theorems generalize results by \textit{K. Bjerklöv, D. Damanik} and \textit{R. Johnson} [Ann. Mat. Pura Appl. (4) 187, No. 1, 1--6 (2008; Zbl 1259.37011)].NEWLINENEWLINEThe authors also remark that considered Schrödinger cocycles play an important role in the study of the spectral problem of the discrete quasi-periodic Schrödinger operator NEWLINE\[NEWLINE [H_f\psi](n)= (\Delta+f(n+(n-1)\alpha))\psi(n) =E \psi (n),NEWLINE\]NEWLINE where \(\Delta \psi(n)= \psi(n+1)+\psi (n-1).\)