Magnetic Schrödinger operators and landscape functions

We study localization properties of low-lying eigenfunctions of magnetic Schrödinger operators $$\frac{1}{2}{(-i\mathrm{\nabla }-A(x))}^2\phi +V(x)\phi =\lambda \phi ,$$ where $V:\mathrm{\Omega }\to {\mathbb{R}}_{\ge 0}$ is a given potential and $A:\mathrm{\Omega }\to {\mathbb{R}}^d$ 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.