Quantitative unique continuation of solutions to the bi‐Laplace equations

2021 ◽  
Author(s):  
Xiaoyu Fu ◽  
Zhonghua Liao
Keyword(s):  
Unique Continuation ◽  
Laplace Equations ◽  
Quantitative Unique Continuation

scholarly journals Band Edge Localization Beyond Regular Floquet Eigenvalues

2020 ◽  
Vol 21 (7) ◽  
pp. 2151-2166
Author(s):  
Albrecht Seelmann ◽  
Matthias Täufer
Keyword(s):  
Floquet Theory ◽  
Large Deviation ◽  
Random Variables ◽  
Unique Continuation ◽  
Band Edge ◽  
Random Schrödinger Operators ◽  
Initial Scale ◽  
Quantitative Unique Continuation ◽  
Scale Estimate ◽  
Floquet Eigenvalues

Abstract We prove that localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in $$L^2({\mathbb {R}}^d)$$ L 2 ( R d ) is universal. By this, we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.

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