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arXiv · 2210.02646

Magnetic Schr\"odinger operators and landscape functions

Jeremy G. Hoskins·Hadrian Quan·Stefan Steinerberger
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Abstract

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.

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Jeremy G. Hoskins, Hadrian Quan, Stefan Steinerberger. 2022-10-06. Magnetic Schr\"odinger operators and landscape functions. https://arxiv.org/abs/2210.02646

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