Mathematics and Computational Sciences

Boundary value problems for the fractional Pauli operator‎: ‎spectral methods and convergence analysis

Document Type : Original Article

Authors

Yusif Gasimov 1, 2
Aynura Aliyeva 3

1 Department of mathematics, Azerbaijan University, Baku, Azerbaijan

2 Department of mathematics, Institute for Physical Problems, Baku State University, Baku, Azerbaijan

3 Department of mathematics, Sumgayit State university, Sumgayit, Azerbaijan

https://doi.org/10.30511/mcs.2025.2074191.1493
Abstract
This paper investigates boundary value problems for the fractional Pauli operator on a finite square domain, addressing a significant gap in the literature where such problems have not been previously studied. The fractional Pauli operator generalizes the standard Pauli operator by replacing the classical Laplacian with the fractional Laplacian (−∆)α=2, introducing non-local quantum effects. We employ
the spectral definition of the fractional Laplacian on bounded domains, expanding the solution as a double trigonometric series that automatically satisfies Dirichlet boundary conditions. The problem is reduced to solving a linear algebraic system for the series coefficients, for which we prove existence and uniqueness in appropriate fractional Sobolev spaces. Numerical experiments for various fractional orders
α demonstrate significant deviations from the classical case (α = 2), with solutions exhibiting enhanced amplitudes and diffusive characteristics as α decreases. Rigorous convergence analysis establishes the continuous transition to the classical Pauli operator as α ! 2−.

Keywords

Fractional Pauli operator
Spectral fractional Laplacian
Boundary value problems
Trigonometric series
Non-local quantum mechanics

Subjects

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Mathematics and Computational Sciences
Volume 7, Issue 1
Winter 2026
Pages 182-191

  • Receive Date 10 October 2025
  • Revise Date 28 October 2025
  • Accept Date 04 November 2025

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