Approximative Analytic Study of Fermions in Magnetar's Crust; Ultra-relativistic Plane Waves, Heun and Mathieu Solutions and Beyond

TL;DR

This paper analyzes fermion behavior in magnetar crusts with periodic magnetic fields, employing perturbative, Heun, and Mathieu solutions to explore wave functions, energy spectra, and mode solutions.

Contribution

It introduces a comprehensive analytic framework combining perturbative, Heun, and Mathieu functions to study fermions in magnetar magnetic fields, extending previous mode analyses.

Findings

Conserved current components computed for ultra-relativistic fermions.

Fermion wave functions expressed via Heun's confluent functions.

Extended analysis of Mathieu's equation solutions and energy spectra.

Abstract

Working with a magnetic field periodic along $Oz$ and decaying in time, we deal with the Dirac-type equation characterizing the fermions evolving in magnetar's crust. For ultra-relativistic particles, one can employ the perturbative approach, to compute the conserved current density components. If the magnetic field is frozen and the magnetar is treated as a stationary object, the fermion's wave function is expressed in terms of the Heun's Confluent functions. Finally, we are extending some previous investigations on the linearly independent fermionic modes solutions to the Mathieu's equation and we discuss the energy spectrum and the Mathieu Characteristic Exponent.