Optimal Spectral Inequality for the Higher-Dimensional Landau Operator

TL;DR

This paper establishes optimal spectral inequalities for Landau operators in any dimension, extending previous results from two dimensions and employing magnetic Bernstein estimates and analyticity techniques.

Contribution

It generalizes spectral inequality results for Landau operators to higher dimensions, overcoming challenges posed by magnetic Bernstein inequalities.

Findings

Proved optimal spectral inequalities in arbitrary dimensions.

Extended results previously known only in 2D.

Implications for control theory, spectral theory, and physics.

Abstract

We prove optimal spectral inequalities for Landau operators in full space and in arbitrary dimension. Spectral inequalities are lower bounds on the L 2 -mass of functions in spectral subspaces of finite energy when integrated over a sampling set S $\subset$ R d . Landau operators are Schr{\"o}dinger operators associated with a constant magnetic field of the form (-$\nabla$ + A(x)) 2 where A is a -in case of non-vanishing magnetic field -unbounded vector potential. Our strategy relies on so-called magnetic Bernstein estimates and analyticity, adapting an approach used by Kovrijkine in the context of the Logvinenko-Sereda theorem. We generalize results previously only known in dimension d = 2. The main difficulty in dimension d $\ge$ 3 are the magnetic Bernstein inequalities which, in comparison to the twodimensional case, lead to additional complications and require more delicate estimates. Our results have immediate consequences for control theory, spectral theory and mathematical physics which we comment on.