6,879 research outputs found
Uniform localization is always uniform
In this note we show that if a family of ergodic Schr\"odinger operators on
with continuous potentials have uniformly localized
eigenfunctions then these eigenfunctions must be uniformly localized in a
homogeneous sense
Discrete Bethe-Sommerfeld Conjecture
In this paper, we prove a discrete version of the Bethe-Sommerfeld
conjecture. Namely, we show that the spectra of multi-dimensional discrete
periodic Schr\"odinger operators on lattice with sufficiently
small potentials contain at most two intervals. Moreover, the spectrum is a
single interval, provided one of the periods is odd, and can have a gap
whenever all periods are even.Comment: 10 page
Generic continuous spectrum for multi-dimensional quasi periodic Schr\"odinger operators with rough potentials
We study the multi-dimensional operator , where is the shift of the torus
\T^d. When , we show the spectrum of is almost surely purely
continuous for a.e. and generic continuous potentials. When ,
the same result holds for frequencies under an explicit arithmetic criterion.
We also show that general multi-dimensional operators with measurable
potentials do not have eigenvalue for generic
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