[Paper Review] On the Aharonov-Casher formula for different self-adjoint extensions of the Pauli operator with singular magnetic field

Mikael Persson|arXiv (Cornell University)|Feb 11, 2005

Spectral Theory in Mathematical Physics16 references22 citations

TL;DR

This paper establishes an Aharonov-Casher-type formula for the maximal self-adjoint extension of the Pauli operator with singular magnetic fields, including Aharonov-Bohm solenoids. It proves that this extension is gauge-invariant and can support both spin-up and spin-down zero modes, unlike previous extensions, and shows it can be approximated by regularized fields, while the Erdős-Vougalter extension cannot.

ABSTRACT

Two different self-adjoint Pauli extensions describing a spin-1/2 two-dimensional quantum system with singular magnetic field are studied. An Aharonov-Casher type formula is proved for the maximal Pauli extension and it is also checked that this extension can be approximated by operators corresponding to more regular magnetic fields.

Motivation & Objective

• To analyze the spectral properties of the Pauli operator with singular magnetic fields, particularly focusing on zero modes.
• To compare two self-adjoint extensions: the maximal Pauli operator and the Erdős-Vougalter (EV) extension.
• To determine whether these extensions can be approximated by regularized magnetic fields via the Borg-Pulé method.
• To investigate the gauge invariance and spectral symmetry of the extensions under magnetic field reversal.
• To derive an Aharonov-Casher-type formula for the maximal Pauli extension and clarify its physical interpretation.

Proposed method

• Defining the Pauli operator via quadratic forms on $ L^2(bR^2) \otimes \bbC^2 $, using vector potentials in $ L^2_{\text{loc}} $.
• Using asymptotic expansions of spinor components near singularities (Aharonov-Bohm solenoids) to characterize domain conditions.
• Applying the condition $ \lim_{\varepsilon \to 0} \varepsilon \int_0^{2\pi} g(\varepsilon e^{i\theta}) \overline{\phi_+(\varepsilon e^{i\theta})} e^{-i\theta} d\theta = 0 $ to determine domain membership.
• Identifying the maximal extension by allowing both $ r^\alpha $ and $ r^{\alpha-1}e^{-i\theta} $ asymptotics in the spin-up component.
• Using gauge transformations to define the EV extension, with flux intensities reduced to $ [-1/2, 1/2) $, breaking full gauge invariance.
• Applying the Borg-Pulé approximation scheme to test whether the extensions arise as limits of regularized magnetic fields.

Experimental results

Research questions

• RQ1 Can an Aharonov-Casher-type formula be established for the maximal self-adjoint extension of the Pauli operator with singular magnetic fields?
• RQ2 Does the maximal Pauli extension support both spin-up and spin-down zero modes simultaneously?
• RQ3 Is the maximal Pauli extension gauge-invariant, and does it transform consistently under magnetic field reversal?
• RQ4 Can the maximal Pauli extension be approximated by regularized magnetic fields in the sense of Borg and Pulé?
• RQ5 How does the EV Pauli extension differ in its spectral properties and approximability compared to the maximal extension?

Key findings

• An Aharonov-Casher-type formula is rigorously proved for the maximal Pauli extension, linking the number of zero modes to the total magnetic flux.
• The maximal Pauli extension supports both spin-up and spin-down zero modes simultaneously, a feature absent in less singular cases.
• The maximal Pauli extension is gauge-invariant and anti-unitarily equivalent under magnetic field reversal, ensuring physical consistency.
• The maximal Pauli extension can be approximated by regularized magnetic fields in the sense of Borg and Pulé, confirming its physical relevance.
• The EV Pauli extension cannot be approximated by regularized fields, and its spectral properties depend on the choice of gauge, breaking invariance under flux sign reversal.
• The parameters $ \nu_0^+ = \infty $, $ \nu_1^+ = 0 $, $ \nu_0^- = 0 $, $ \nu_1^- = \infty $ characterize the maximal extension, indicating specific asymptotic behavior of spinor components.