Multiscale unique continuation properties of eigenfunctions

TL;DR

This paper reviews and introduces new scale-free quantitative unique continuation principles for multiscale Schr"odinger and elliptic operators, with applications in control theory and random operators, emphasizing explicit estimates and Carleman techniques.

Contribution

It presents new scale-free unique continuation estimates for PDEs with multiscale structures, including explicit Carleman estimates, advancing theoretical tools for spectral and control applications.

Findings

Scale-free unique continuation estimates for multiscale PDEs

Explicit Carleman estimate for second order elliptic operators

Applications to spectral projections and control theory

Abstract

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.