Magnetic Schrödinger operators and landscape functions

10/06/2022

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by   Jeremy G. Hoskins, et al.

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We study localization properties of low-lying eigenfunctions of magnetic Schrödinger operators 1/2(- i∇ - A(x))^2 ϕ + V(x) ϕ = λϕ, where V:Ω→ℝ_≥ 0 is a given potential and A:Ω→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.