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We consider Schrödinger operators on the real line with limit-periodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we show that for a dense set of limit-periodic potentials, the spectrum of the associated Schrödinger operator has Hausdorff dimension zero. In both results one can introduce a coupling constant , and the respective statement then holds simultaneously for all values of the coupling constant.
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David Damanik, Jake Fillman, Milivoje Lukic, Limit-periodic continuum Schrödinger operators with zero measure Cantor spectrum. J. Spectr. Theory 7 (2017), no. 4, pp. 1101–1118DOI 10.4171/JST/186