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

Multiscale unique continuation properties of eigenfunctions