8,345 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
Shnol's theorem and the spectrum of long range operators
We extend some basic results known for finite range operators to long range
operators with off-diagonal decay. Namely, we prove an analogy of Sch'nol's
theorem. We also establish the connection between the almost sure spectrum of
long range random operators and the spectra of deterministic periodic
operators.Comment: To appear in Proc. AMS. Referee's comments incorporate
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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