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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
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