Magnetic Bernstein inequalities and spectral inequality on thick sets for the Landau operator
Paul Pfeiffer, Matthias Täufer
TL;DR
The paper establishes a sharp, energy-dependent spectral inequality for the 2D Landau operator on thick sets, proven via magnetic Bernstein inequalities and an analyticity-based local bound. It provides explicit constants showing how the observation set geometry, the magnetic field, and the energy scale combine to control Fourier-analytic mass, and extends the results to finite domains. These inequalities yield null-controllability for the magnetic heat equation with sharp cost and improve Wegner estimates and IDS regularity for continuum random Schrödinger operators, even when the single-site potential is only positive on a thick set. Collectively, the work broadens quantitative unique continuation for the Landau operator and relaxes openness assumptions in random media, with concrete implications for control theory and localization phenomena.
Abstract
We prove a spectral inequality for the Landau operator. This means that for all $f$ in the spectral subspace corresponding to energies up to $E$, the $L^2$-integral over suitable $S \subset \mathbb{R}^2$ can be lower bounded by an explicit constant times the $L^2$-norm of $f$ itself. We identify the class of all measurable sets $S \subset \mathbb{R}^2$ for which such an inequality can hold, namely so-called thick or relatively dense sets, and deduce an asymptotically optimal expression for the constant in terms of the energy, the magnetic field strength and in terms of parameters determining the thick set $S$. Our proofs rely on so-called magnetic Bernstein inequalities. As a consequence, we obtain the first proof of null-controllability for the magnetic heat equation (with sharp bound on the control cost), and can relax assumptions in existing proofs of Anderson localization in the continuum alloy-type model.