Magnetic Schrödinger operators and landscape functions

We study localization properties of low-lying eigenfunctions of magnetic Schr\"odinger operators $$\frac{1}{2} \left(- i\nabla - A(x)\right)² \phi + V(x) \phi = \lambda \phi,$$ where $V:\Omega \rightarrow \mathbb{R}_{\geq 0}$ is a given potential and $A:\Omega \rightarrow \mathbb{R}^d$ induces a magnetic field. We extend the Filoche-Mayboroda inequality and prove a refined inequality in the magnetic setting which can predict the points where low-energy eigenfunctions are localized. This result is new even in the case of vanishing magnetic field. Numerical examples illustrate the results.