## Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions.(English)Zbl 0990.39014

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 $$C$$.

### MSC:

 39A12 Discrete version of topics in analysis 39A70 Difference operators 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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