Absence of positive eigenvalues of magnetic Schr\"odinger operators

TL;DR

This paper establishes conditions under which magnetic Schr"odinger operators in multi-dimensional spaces lack positive eigenvalues, with implications for related quantum operators like Pauli, Dirac, and Aharonov--Bohm.

Contribution

It provides a sharp criterion for the absence of positive eigenvalues based on magnetic field decay, extending understanding of spectral properties of magnetic Schr"odinger operators.

Findings

Eigenvalues above a certain energy threshold are absent.

The results are sharp regarding magnetic field decay.

Applications include spectral analysis of Pauli, Dirac, and Aharonov--Bohm operators.

Abstract

We study sufficient conditions for the absence of positive eigenvalues of magnetic Schr\"odinger operators in $\mathbb{R}^d,\, d\geq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller--Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov--Bohm operators.