Fock Spaces, Landau Operators and the Regular Solutions of time-harmonic Maxwell equations

TL;DR

This paper combines hypercomplex analysis and quantum mechanics to characterize solutions of time-harmonic Maxwell equations using series expansions with spherical harmonics and Landau operator eigenfunctions, impacting quantum and relativistic physics.

Contribution

It introduces a novel representation of Maxwell solutions through series involving spherical harmonics and Landau operators, linking electromagnetic and quantum mechanical frameworks.

Findings

Series expansions of solutions in terms of spherical harmonics and monogenics.

Eigenfunction analysis of Landau operators for solution representation.

Implications for regularity and hypo-ellipticity in quantum mechanics.

Abstract

We investigate the representations of the solutions to Maxwell's equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the solutions to the time-harmonic Maxwell system in terms of series expansions involving spherical harmonics resp. spherical monogenics. Also, a thorough investigation for the series representation of the solutions in terms of eigenfunctions of Landau operators that encode $n-$dimensional spinless electrons is given. This new insight should lead to important investigations in the study of regularity and hypo-ellipticity of the solutions to Schr\"odinger equations with natural applications in relativistic quantum mechanics concerning massive spinor fields.