## Almost periodic Schrödinger operators. II: The integrated density of states.(English)Zbl 0544.35030

[For part I, see Commun. Math. Phys. 82, 101-120 (1981; Zbl 0484.35069).]
The authors study Schrödinger operators with almost periodic potential $$V$$ on $$\mathbb R^m$$, and, for the one-dimensional case, $$m=1$$, also its finite difference analogue. The Lyapunov index $$\gamma(E)$$ and the density of states $$k(E)$$ are investigated. It is shown, that the spectrum of the Schrödinger operator coincides with the growth points of $$k$$. Furthermore a formula derived by D. Thouless [J. Phys. C 5, 77–81 (1972)], in the context of random potentials $\gamma(E)=\int^{\infty}_{-\infty}\log | E-E'| \quad d(k-k_ 0)(E')+\gamma_ 0(E)$ $$(k_ 0,\gamma_ 0$$ explicitly known constants) which connects the two objects is proven. For a special almost periodic function, $$V(n)=\lambda \cos(2\pi \alpha n),$$ the following symmetry relation known as Aubry duality $$k(\alpha,\lambda,E)=k(\alpha,4/\lambda,2E/\lambda)$$ is proven. Finally, a Jacobi matrix and, by a theorem of Bellisard and Simon, a Schrödinger operator with singular continuous spectrum is constructed.
Reviewer: H.Siedentop

### MSC:

 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 47B39 Linear difference operators 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35P05 General topics in linear spectral theory for PDEs

Zbl 0484.35069
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### References:

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