Magnetic Schrödinger operators and landscape functions

2022/10/06 by Jeremy Hoskins, Hoskins, Jeremy G., Hadrian Quan +3
Mathematics · #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2210.02646

Abstract

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.

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