Consider the discrete quasi-periodic Schrödinger equation at energy \(E\): \[ -\psi_{n+1}- \psi_{n-1}+ V(\theta+ n\omega)\psi_n= E\psi_{n-1}, \] where the potential \(V:\mathbb{T}^d \to\mathbb{T}^d\) is a real-valued function and \(\theta \mapsto\theta +\omega\) denotes ergodic shift. Then \[ M_n(\theta, E)= \prod^n_{j=1} A(\theta+j \omega,E) \] is the monodromy matrix and \[ L_n(E)= {1 \over n}\int_\mathbb{T} \log\bigl \|M_n(x,E) \bigr\|dx \] is the Lyapunov exponent. \[ \text{If }L(E)= \int\log |E-E'|dN(E') \] then \(N(\cdot)\) is called integrated density of states. The paper is devoted to the study of regularity properties of \(L(E)\) and \(N(E)\). Among others:
– If \(L(E)> \gamma>0\) for \(E\in I\) \((I\) is a compact interval), then \(0\leq L_n(E)-L(E)\leq C_0 /n\), \(n=1,2, \dots\) with some \(C_0\);
– there exists \(\beta\) such that \(|L(E)-L(E') |+ |N(E)-N(E') |\leq C|E-E' |^\beta\) for all \(E\), \(E'\in I\) and some \(C\).