Spectral Properties of the Fractional Pauli Operator on a Bounded Domain

Gasimov, Yusif; Aliyeva, Aynura

doi:10.30473/coam.2026.76342.1354

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	Spectral Properties of the Fractional Pauli Operator on a Bounded Domain
Control and Optimization in Applied Mathematics
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 08 بهمن 1404 اصل مقاله (472.87 K)
نوع مقاله: Research Article
شناسه دیجیتال (DOI): 10.30473/coam.2026.76342.1354
نویسندگان
Yusif Gasimov^* ¹^{، 2}^{، 3}؛ Aynura Aliyeva⁴
¹Azerbaijan University, Baku, Azerbaijan
²Institute for Physical Problems, Baku State University, Baku, Azerbaijan
³Institute of Mathematics, Ministry of Science and Education, Baku, Azerbaijan
⁴Sumgayit State University, Sumgayit, Azerbaijan
چکیده
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.
تازه های تحقیق
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.
کلیدواژه‌ها
Fractional Pauli operator؛ Quantum mechanics؛ Eigenvalue problem؛ Discrete spectrum

مراجع
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Spectral Properties of the Fractional Pauli Operator on a Bounded Domain