Approximative analytic study of fermions in magnetar’s crust; ultra-relativistic plane waves, Heun and Mathieu solutions and beyond

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.

Analytic Results in the Position-Dependent Mass Schrödinger Problem

“…Interestingly enough, for a long time absent in the literature, Heun functions have recently been found in very different contexts, see e.g. [57][58][59][60][61][62]. Besides exactly obtaining all the solutions in the three cases studied we have plotted all the first eigenstates in a systematic way, emphasizing their parity properties.…”

Energy eigenfunctions for position-dependent mass particles in a new class of molecular Hamiltonians

“…Heun functions have recently been receiving increasing attention and have been found in a wide variety of contexts. [83][84][85][86][87][88] PDM particles tend to be more likely tunneling than ordinary ones. Next, we have addressed the PDM version of the sixth power hyperbolic potential V (x) = −Asech 6 x − Bsech 4 x and obtained exact expressions for the zero-modes in the A and A = − B cases belonging to a discrete set of parameters.…”

“…We have analytically found confluent-Heun expressions in the general case and compared the PDM eigenfunctions to the constant-mass ones. Heun functions have recently been receiving increasing attention and have been found in a wide variety of contexts [83][84][85][86][87][88]. PDM particles tend to be more likely tunneling than ordinary…”

Solutions to position-dependent mass quantum mechanics for a new class of hyperbolic potentials

“…In the context of the ordinary constant mass Schrödinger equation, the author of [42] signaled the role of the confluent Heun functions in the dynamics of two family members, emphasizing the fact that Heun functions are rather exotic though increasingly popular in contemporary physics and mathematics. Indeed, Heun functions have been practically absent in the literature for almost one century but have been lastly emerging in all frontiers of physics [66][67][68][69][70][71]. In the present paper, we reconsidered the same class of potentials when the mass is promoted to a phenomenologically interesting soliton-like distribution.…”