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 $L\in 2\NN+1$. 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 $\delta >0$ 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.

This file consist of two parts. The first (including Appendix A) is the final manuscript submitted in September '12 to appear in CMP. Appendix B is part of the first version of February '12. It concerns an estimate on local mass fluctuations inside dominating boxes, a result which is not used in the main body of the paper, but may be of independent interest

References in corpus (3)

• Bounds on the density of states for Schr\" odinger operators
• Characterization of the Anderson metal-insulator transition for non ergodic operators and application
• A linear Wegner estimate for alloy type Schroedinger operators on metric graphs

Cited by in corpus (22)

• An abstract Logvinenko-Sereda type theorem for spectral subspaces
• Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus
• Ergodicity and dynamical localization for Delone-Anderson operators
• Scale-free uncertainty principles and Wegner estimates for random breather potentials
• Semiclassical resolvent estimates for bounded potentials
• Conditional Wegner estimate for the standard random breather potential
• Band edge localization beyond regular Floquet eigenvalues
• A quantitative Carleman estimate for second order elliptic operators
• Wegner estimate and disorder dependence for alloy-type Hamiltonians with bounded magnetic potential
• Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations
• Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators
• Wegner estimate for Landau-breather Hamiltonians
• Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials
• Localisation for Delone operators via Bernoulli randomisation
• A lower Wegner estimate and bounds on the spectral shift function for continuum random Schrödinger operators
• A bound on the averaged spectral shift function and a lower bound on the density of states for random Schrödinger operators on ${\mathbb{R}}^d$
• Some abstract Wegner estimates with applications
• Random Schrödinger Operators and Anderson localization in aperiodic media
• Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
• Wegner estimate for random divergence-type operators monotone in the randomness
• Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators
• Sampling inequality for $L^2$-norms of eigenfunctions, spectral projectors, and Weyl sequences of Schrödinger operators