Magnetic Bloch eigenvalue problems in nonuniform periodic magnetic fields: a Fourier-finite difference approach
Abstract
We study magnetic Bloch eigenvalue problems for two-dimensional magnetic Schrödinger operators with nonuniform periodic magnetic fields and periodic scalar potentials, and develop a hybrid Fourier-finite difference discretization. A Landau-type gauge is constructed by solving a periodic Poisson equation. Under the integer magnetic flux condition, the magnetic Bloch boundary conditions are incorporated through shifts of Fourier-mode indices. The method combines link-variable-based covariant central differences in the x-direction with a Fourier pseudospectral discretization in the y-direction, yielding a Hermitian discrete Hamiltonian. The resulting eigenvalue problems are solved using a LOBPCG method. Under appropriate regularity assumptions, we establish a fully discrete consistency residual estimate of order O(h12+N2−s). With suitable spectral separation and eigenspace dimension-preserving conditions, error estimates of the same order are derived for isolated eigenvalues and their associated eigenspaces. Numerical experiments show second-order convergence in the finite difference direction and rapid error reduction with increasing Fourier resolution, and illustrate band degeneracies and eigenstate distributions in nonuniform periodic magnetic fields.
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BibTeX
@misc{li2026magnetic,
title = {{Magnetic Bloch eigenvalue problems in nonuniform periodic magnetic fields: a Fourier-finite difference approach}},
author = {Xueyang Li and Yuanwei Lai},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05950},
eprint = {2611.05950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05950},
}
RIS
TY - UNPB
TI - Magnetic Bloch eigenvalue problems in nonuniform periodic magnetic fields: a Fourier-finite difference approach
AU - Xueyang Li
AU - Yuanwei Lai
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05950
UR - https://arxiv.org/abs/2611.05950
AB - We study magnetic Bloch eigenvalue problems for two-dimensional magnetic Schrödinger operators with nonuniform periodic magnetic fields and periodic scalar potentials, and develop a hybrid Fourier-finite difference discretization. A Landau-type gauge is constructed by solving a periodic Poisson equation. Under the integer magnetic flux condition, the magnetic Bloch boundary conditions are incorporated through shifts of Fourier-mode indices. The method combines link-variable-based covariant central differences in the x-direction with a Fourier pseudospectral discretization in the y-direction, yielding a Hermitian discrete Hamiltonian. The resulting eigenvalue problems are solved using a LOBPCG method. Under appropriate regularity assumptions, we establish a fully discrete consistency residual estimate of order O(h_1^2+N_2^−s). With suitable spectral separation and eigenspace dimension-preserving conditions, error estimates of the same order are derived for isolated eigenvalues and their associated eigenspaces. Numerical experiments show second-order convergence in the finite difference direction and rapid error reduction with increasing Fourier resolution, and illustrate band degeneracies and eigenstate distributions in nonuniform periodic magnetic fields.
ER -