Magnetic Schrödinger operators and landscape functions

Abstract

We study localization properties of low-lying eigenfunctions of magnetic Schrödinger operators ${(-i\nabla -A(x))}^2\varphi +V(x)\varphi =\lambda \varphi ,$ where $V:\Omega \to {\mathbb{R}}_{\ge 0}$ is a given potential and $A:\Omega \to {\mathbb{R}}^d$ 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 $A\equiv 0$. Numerical examples illustrate the results.

Keywords:

• Localization
• eigenfunction
• Schrödinger operator
• regularization
• Primary: 35J10, 65N25
• Secondary: 82B44

Additional information

Funding

S.S. is supported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation.