On the eigenvalues of the fermionic angular eigenfunctions in the Kerr metric

TL;DR

This paper investigates the eigenvalues of fermionic angular functions in the Kerr metric, deriving a PDE for these eigenvalues, proving the impossibility of certain ODE formulations, and developing perturbative and asymptotic expansions.

Contribution

It introduces a new PDE approach for angular eigenvalues, proves limitations on ODE formulations, and provides novel perturbative and asymptotic expansions for Kerr eigenvalues.

Findings

Derived a quasi-linear PDE for angular eigenvalues

Proved no ODE in particle energy or black hole mass exists for eigenvalues

Constructed perturbative expansions and asymptotic formulas for Kerr eigenvalues

Abstract

In view of a result recently published in the context of deformation theory of linear Hamiltonian systems, we reconsider the eigenvalue problem associated to the angular equation arising after the separation of the Dirac equation in the Kerr metric and we show how efficiently a quasi-linear first order PDE for the angular eigenvalues can be derived. We also prove that it is not possible to obtain an ordinary differential equation for the eigenvalues where the role of the independent variable is played by the particle energy or the black hole mass. Finally, we construct new perturbative expansions for the eigenvalues in the Kerr case and obtain an asymptotic formula for the eigenvalues in the case of a Kerr naked singularity.