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The spectrum of a quasiperiodic Schrödinger operator. (English) Zbl 0624.34017

The spectrum \(\sigma\) (H) of the tight binding Fibonacci Hamiltonian \((H_{mn}=\delta_{m,n+1}+\delta_{m+1,n}+\delta_{m,n}\mu v(n)\), \(v(n)=\chi_{[-\omega^ 3,\omega^ 2[}((n-1)\omega)\), 1/\(\omega\) is the golden number) is shown to coincide with the dynamical spectrum, the set on which an infinite subsequence of traces of transfer matrices is bounded. The point spectrum is absent for any \(\mu\), and \(\sigma\) (H) is a Cantor set for \(| \mu | \geq 4\). Combining this with Casdagli’s earlier result, one finds that the spectrum is singular continuous for \(| \mu | \geq 16\).

MSC:

34L99 Ordinary differential operators
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