Construction of QuasiPeriodic Schrödinger Operators with Cantor Spectrum

Abstract

It is well known that, for fixed Diophantine frequencies and generic small smooth or analytic quasiperiodic potentials, both continuous and discrete Schrodinger operators have Cantor spectrum. Although there have been several examples of Schrodinger operators with Cantor spectrum since Moser’s pioneering work (Bellissard, in: Luck, Moussa, Waldschmidt (eds) Number theory and physics (Les Houches, 1989), Springer, Berlin, 1990; Bellissard et al. in Phys Rev Lett 49:701–704, 1982; Damanik et al. in Ann Henri Poincare 15:1123–1144, 2014, J Spectr Theory 7:1101–1118, 2017; Moser in Comment Math Helv 56:198–224, 1981); however, so far there is no concrete quasiperiodic example in the continuous case, and there is no concrete quasiperiodic example in the discrete case besides the cosine-like potentials (Avila and Jitomirskaya in Ann Math 170:303–342, 2009; Puig in Commun Math Phys 244:297–309, 2004; Sinai in J Stat Phys 46:861–909; 1987; Wang and Zhang in Int Math Res Not 2017:2300–2336, 2017). In this paper, we present a strategy for explicitly constructing quasiperiodic Schrodinger operators with Cantor spectrum.