Magnetic Schrodinger operators and landscape functions

We study localization properties of low-lying eigenfunctions of magnetic Schrodinger operators(-i backward difference -A(x))2 phi+V(x)phi=lambda phi,where V:omega -> R >= 0 is a given potential and A:omega -> Rd 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 A equivalent to 0 . Numerical examples illustrate the results.