Uniform Cantor Singular Continuous Spectrum for Nonprimitive Schrödinger Operators
نویسنده
چکیده
It is shown that some Schrödinger operators, with nonprimitive substitution potentials, have pure singular continuous Cantor spectrum with null Lebesgue measure for all elements in the respective hulls.
منابع مشابه
Limit-periodic Schrödinger Operators in the Regime of Positive Lyapunov Exponents
We investigate the spectral properties of the discrete one-dimensional Schrödinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a Cantor group. We show that for given Cantor group and minimal translation, there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero Hausdorff dime...
متن کاملLimit-periodic Continuum Schrödinger Operators with Zero Measure Cantor Spectrum
We consider Schrödinger operators on the real line with limitperiodic 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 i...
متن کاملSingular Continuous Spectrum for Certain Quasicrystal Schrödinger Operators
We give a short introduction into the theory of one-dimensional discrete Schrödinger operators associated to quasicrystals. We then report on recent results, obtained in jont work with D. Damanik, concerning a special class of these operators viz Quasi-Sturmian operators. These results show, in particular, uniform singular continuous spectrum of Lebesgue measure zero.
متن کاملSpectral Properties of Limit-periodic Schrödinger Operators
We investigate the spectral properties of Schrödinger operators in `2(Z) with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show tha...
متن کاملUniform Spectral Properties of One-dimensional Quasicrystals, Iv. Quasi-sturmian Potentials
We consider discrete one-dimensional Schrr odinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dy-namical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a C...
متن کامل