Frequency dependence of Hölder continuity for quasiperiodic Schrödinger operators

Abstract

We prove estimates on the Hölder exponent of the density of states measure for discrete Schrödinger operators with potential of the form V (n) = λ ( (n+1)β − nβ ), with λ large enough, and conclude that for almost all values of β , the density of states measure is not Hölder continuous.

The Spectral Properties of the Strongly Coupled Sturm Hamiltonian of Eventually Constant Type

“…This generalizes the results in [12]. Munger [34] considered the frequency of constant type, i.e. α κ = [κ; κ, κ, · · · ].…”

“…(3) For α = α κ , the first equality in (iv) is (1.5) ( see [34]); for any irrational α, the third equality in (iv) is (1.4) (see [27,19,28]). We state them here for comparison.…”

“…Acknowledgements. The author would like to thank professor Anton Gorodetski for pointing out the references [16,23,32,34], which are closely related to this paper. He also thanks professor Jean Bellissard for pointing out that the results actually hold for more general frequencies, namely, frequencies of eventually constant type and encouraging me to write down this new version, which greatly improves the previous version of the paper.…”

Exact-Dimensional Property of Density of States Measure of Sturm Hamiltonian

“…This generalizes the results in [7]. Munger [24] gave estimation on the optimal Hölder exponent of N ακ,λ and obtained asymptotic formula for it. For λ > 20, Qu [27] obtained the dimension formula of N ακ,λ similar with (1.1), he also showed that N ακ,λ is exact-dimensional and obtained similar asymptotic behavior as (1.2).…”

Schrödinger operators with dynamically defined potentials

“…and the lower densityd * (α) := lim inf K→∞ 1 K K k=1 a k < ∞ of α.Namely, still assuming λ > 24, Munger[183] has also shown that N λ,α is Hölder continuous if d * (α) is finite, and it is not Hölder continuous if d * (α) is infinite. Transport Exponents.…”