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Zero measure Cantor spectra for continuum one-dimensional quasicrystals. (English) Zbl 1351.47030

In this paper, the authors consider Schrödinger operators on \(\mathbb{R}\) with measures as potentials. Choosing a suitable subset of measures, they work with a dynamical system consisting of measures. Then, they relate properties of this dynamical system with spectral properties of the associated operators, and with the obtained result they prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated by aperiodic subshifts.

MSC:

47B80 Random linear operators
47A10 Spectrum, resolvent
47A35 Ergodic theory of linear operators
35J10 Schrödinger operator, Schrödinger equation
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