Document Type : Research Article

Authors

• Yusif Gasimov ^{1, 2}
• Aynura Aliyeva ³

¹ Azerbaijan University, Baku, Azerbaijan

² Institute for Physical Problems, Baku State University, Baku, Azerbaijan

³ Sumgayit State University, Sumgayit, Azerbaijan

Abstract

 This paper introduces and analyzes, for the first time, the \emph{fractional Pauli operator}, a non-local generalization of the fundamental quantum mechanical operator describing spin-1/2 particles in magnetic fields. The operator is defined through the spectral theory of the magnetic fractional Laplacian $(H_{\vecA})^s$, with s ∈ (0,1), and acts on spinor-valued wavefunctions. We formulate the associated eigenvalue problem on a bounded domain Ω ⊂ ℝ^2 subject to exterior Dirichlet conditions. The intrinsic non-locality of the model is addressed via a variational formulation in suitable magnetic fractional Sobolev spaces. Under appropriate assumptions on the vector potential $\vecA$ and the magnetic field B, we establish the existence of a discrete spectrum. For a constant magnetic field on \R^2, we derive explicit eigenvalues exhibiting a nonlinear B_0^s scaling of the Landau levels. In addition, a finite element–based numerical scheme is developed to compute the spectrum on a disk, illustrating the combined effects of spatial confinement and non-locality. The physical implications of fractional kinetic effects on Landau quantization and spin-dependent phenomena are discussed, highlighting the relevance of the fractional Pauli operator for modeling anomalous transport in bounded quantum systems.

Highlights

• Introduction of the fractional Pauli operator Ps as a non-local, spectral generalization of the Pauli operator for spin-1/2 particles in magnetic fields, built on the magnetic fractional Laplacian (H_A) ^s with s ∈ (0,1).
• Well-posed eigenvalue problem formulated on bounded domains Ω ⊂ ℝ^2 with exterior Dirichlet-like conditions, establishing existence and discreteness of the spectrum under suitable assumptions on A and B.
• Explicit Landau-type spectrum derived for a constant magnetic field on ℝ^2, revealing a nonlinear B_0^s scaling of Landau levels unlike the classical linear scaling.
• Rich interplay between spin, non-local fractional kinetics, and geometric confinement, showing domain shape and boundary conditions influence spectral properties and Landau quantization.
• Finite element–based numerical framework developed to compute the spectrum on realistic domains (e.g., a disk), demonstrating practical computability of fractional Pauli spectra in confined geometries.
• Physical implications discussed: modified Landau quantization and spin-dependent phenomena due to fractional dynamics, with potential relevance to anomalous transport and quantum Hall-like behavior in systems with anomalous diffusion.

Keywords

• Fractional Pauli operator
• Quantum mechanics
• Eigenvalue problem
• Discrete spectrum

Main Subjects

• Applied & Interdisciplinary