91 research outputs found
Discrete alloy-type models: Regularity of distributions and recent results
We consider discrete random Schr\"odinger operators on with a potential of discrete alloy-type structure. That is, the
potential at lattice site is given by a linear combination
of independent identically distributed random variables, possibly with
sign-changing coefficients. In a first part we show that the discrete
alloy-type model is not uniformly -H\"older continuous, a frequently used
condition in the literature of Anderson-type models with general random
potentials. In a second part we review recent results on regularity properties
of spectral data and localization properties for the discrete alloy-type model.Comment: 20 pages, 0 figure
Equidistribution estimates for eigenfunctions and eigenvalue bounds for random operators
We discuss properties of -eigenfunctions of Schr\"odinger operators and
elliptic partial differential operators. The focus is set on unique
continuation principles and equidistribution properties. We review recent
results and announce new ones.Comment: Keywords: scale-free unique continuation property, equidistribution
property, observability estimate, uncertainty relation, Carleman estimate,
Schr\"odinger operator, elliptic differential equatio
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