Optimal Spectral Inequality for the Higher-Dimensional Landau Operator
Sedef Özcan, Matthias Täufer
TL;DR
The paper proves an optimal, dimension‑dependent spectral inequality for the magnetic Landau operator $H_B$ on $\mathbb{R}^d$, $d\ge 3$, with thick sampling sets $S$. It combines magnetic Bernstein estimates with analytic continuation in Kovrijkine’s framework to obtain explicit observability constants that depend on the energy $E$, thick-set geometry $(\ell,\rho)$, and the magnetic field $B$, and shows the $|\ell|_1$-dependence in the exponent is sharp. The authors also establish a detailed algebra of magnetic derivatives, derive high-order bounds via polynomial control of $R^{m}(\mathrm{Id})$ by powers of $H_B$, and develop an analytic foundation that includes dimension reduction and a good/bad rectangle decomposition to prove the main inequality. This result extends the known $d=2$ Landau operator case to higher dimensions and has immediate implications for null-controllability of magnetic heat equations, semiclassical eigenvalue estimates, and Anderson localization in magnetic Schrödinger settings. The methods provide a robust framework for quantitative unique continuation in the presence of constant magnetic fields in full space.
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{ö}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.