Magnetic Bernstein inequalities and spectral inequality on thick sets for the Landau operator

TL;DR

This paper establishes spectral inequalities for the Landau operator on thick sets, deriving explicit bounds and applying results to control theory and localization phenomena in quantum physics.

Contribution

It introduces magnetic Bernstein inequalities and characterizes sets for which spectral inequalities hold, advancing understanding of spectral properties of the Landau operator.

Findings

Spectral inequality for Landau operator on thick sets

Explicit bounds depending on energy and magnetic field

First proof of null-controllability for magnetic heat equation

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.