Scale-Free Unique Continuation Estimates and Applications to Random Schrödinger Operators

Abstract

We prove a unique continuation principle or uncertainty relation valid for Schrödinger operator eigenfunctions, or more generally solutions of a Schrödinger inequality, on cubes of side L ∈ 2N + 1. It establishes an equi-distribution property of the eigenfunction over the box: the total L 2 -mass in the box of side L is estimated from above by a constant times the sum of the L 2 -masses on small balls of a fixed radius δ > 0 evenly distributed throughout the box. The dependence of the constant on the various parameters entering the problem is given explicitly. Most importantly, there is no L-dependence.This result has important consequences for the perturbation theory of eigenvalues of Schrödinger operators, in particular random ones. For so-called Delone-Anderson models we deduce Wegner estimates, a lower bound for the shift of the spectral minimum, and an uncertainty relation for spectral projectors.

Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

“…Our main theorem unifies and generalizes all the results mentioned so far and makes the dependence on the model parameters explicit. Indeed, our scale-free unique continuation principle answers positively a question asked in [40]. A partial answer was given already in [27].…”

“…A partial answer was given already in [27]. While [40] concerns the case of a single eigenfunctions, [27] treats linear combinations of eigenfunctions corresponding to very close eigenvalues. For a broader discussion we refer to the summer school notes [43].…”

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

“…This allowed the dropping of the ln term in (3). A full answer to the question raised in [17] was given by the following Theorem.…”

Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators

“…More than this, there are several quantitative formulations of unique continuation which proved to be useful in a variety of applications, see e.g. [BK05,RL12,BK13,RMV13,NTTV16]. For instance, Bourgain and Kenig [BK05] showed that if ∆u = V u in R d , u(0) = 1 and u, V ∈ L ∞ (R d ) then for all x ∈ R d with |x| > 1 we have max |y−x|≤1 |u(y)| > c · exp −c ′ (log|x|)|x| 4/3 .…”

“…For the application to random Schrödinger operators it is crucial that the result is scale-free, i.e. C is independent of L. In [RMV13] the question was raised whether a similar estimate holds for finite linear combinations of…”