Showing papers on "Almost Mathieu operator published in 2000"

Hölder Regularity of Integrated Density of States for the Almost Mathieu Operator in a Perturbative Regime

Abstract: Consider the Almost Mathieu operator Hλ= λ cos 2π(kω +θ)+Δ on the lattice. It is shown that for large λ, the integrated density of states is Holder continuous of exponent κ < \(\frac{1}{2}\). This result gives a precise version in the perturbative regime of recent work by M. Goldstein and W. Schlag on Holder regularity of the integrated density of states for 1D quasi-periodic lattice Schrodinger operators, assuming positivity of the Lyapunov exponent (and proven by different means). Our approach provides also a new way to control Green's functions, in the spirit of the author's work in KAM theory. It is by no means restricted to the cosine-potential and extends to band operators.

Non-invertibility of certain almost Mathieu operators

Abstract: It is shown that the almost Mathieu operators of the type Te n =e n-1 + λsin(2nr)e n + e n+1 where λ is real and r is a rational multiple of π and {e n :n = 1,2,3,....} an orthonormal basis for a Hilbert space, is notinvertible.

Regularity of the spectrum for the almost Mathieu operator

Abstract: It is shown that the logarithmic potential associated with the integrated density of states is constant on the spectrum of the almost Mathieu operator in case the irrational frequency is sufficiently well approximable by rationals in terms of a diophantine condition.