Magnetic Schr\"odinger operators and landscape functions

TL;DR

This paper investigates how magnetic fields influence the localization of low-energy eigenfunctions of Schrödinger operators, extending existing inequalities to predict localization points even without magnetic fields, supported by numerical examples.

Contribution

It extends the Filoche-Mayboroda inequality to magnetic Schrödinger operators, providing a refined inequality that predicts eigenfunction localization points.

Findings

Refined inequality predicts localization points in magnetic settings.

Results are applicable even when magnetic field is absent.

Numerical examples validate theoretical predictions.

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.