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

Abstract

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector ${\chi }_{(-\infty ,E]}(H_L)$ of a Schrödinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. We apply it to (i) prove a Wegner estimate for random Schrödinger operators with non-linear parameter-dependence and to (ii) exhibit the dependence of the control cost on geometric model parameters for the heat equation in a multi-scale domain.

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Cited by in corpus (15)

• Sharp geometric condition for null-controllability of the heat equation on ${\mathbb{R}}^d$ and consistent estimates on the control cost
• An abstract Logvinenko-Sereda type theorem for spectral subspaces
• Sufficient criteria and sharp geometric conditions for observability in Banach spaces
• Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus
• Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller
• Semiclassical resolvent estimates for bounded potentials
• Band edge localization beyond regular Floquet eigenvalues
• 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
• Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials
• Lifshitz tails for Schrödinger operators with random breather potential
• A lower Wegner estimate and bounds on the spectral shift function for continuum random Schrödinger operators
• Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators
• Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
• Wegner estimate for random divergence-type operators monotone in the randomness