Spectral properties and time decay of the wave functions of Pauli and Dirac operators in dimension two

TL;DR

This paper analyzes how magnetic fields affect the spectral properties and time decay of wave functions for two-dimensional Pauli and Dirac operators, revealing the role of magnetic flux in resolvent expansions and wave decay behavior.

Contribution

It provides explicit resolvent expansions near thresholds for these operators, linking the asymptotics of zero modes to magnetic flux and computing singular terms explicitly.

Findings

Resolvent expansions are determined by magnetic flux.

Zero mode asymptotics are matched via two methods.

Magnetic field influences wave-function decay, illustrating paramagnetic and diamagnetic effects.

Abstract

We consider two-dimensional Pauli and Dirac operators with a polynomially vanishing magnetic field. The main results of the paper provide resolvent expansions of these operators in the vicinity of their thresholds. It is proved that the nature of these expansions is fully determined by the flux of the magnetic field. The most important novelty of the proof is a comparison between the spatial asymptotics of the zero modes obtained in two different manners. The result of this matching allows to compute explicitly all the singular terms in the associated resolvent expansions. As an application we show how the magnetic field influences the time decay of the associated wave-functions quantifying thereby the paramagnetic and diamagnetic effects of the spin.