  Quantitative unique continuation principles for multiscale structures are an
important ingredient in a number applications, e.g. random Schr\"odinger
operators and control theory.
  We review recent results and announce new ones regarding quantitative unique
continuation principles for partial differential equations with an underlying
multiscale structure. They concern Schr\"odinger and second order elliptic
operators. An important feature is that the estimates are scale free and with
quantitative dependence on parameters. These unique continuation principles
apply to functions satisfying certain `rigidity' conditions, namely that they
are solutions of the corresponding elliptic equations, or projections on
spectral subspaces. Carleman estimates play an important role in the proofs of
these results. We also present an explicit Carleman estimate for second order
elliptic operators.
