On the Eigenvalues of the Chandrasekhar-Page Angular Equation

TL;DR

This paper analyzes the eigenvalues of the Chandrasekhar-Page angular equation, revealing their dependence on parameters, their analytic properties, and connections to Painleve III equations, with implications for black hole physics.

Contribution

The paper introduces a self-adjoint operator family for the eigenvalue problem, proves eigenvalue analyticity, derives a PDE and power series expansions, and explores the eigenvalues' characterization via a holomorphic function.

Findings

Eigenvalues depend analytically on parameters  and .

Eigenvalues satisfy a PDE reducible to Painleve III.

Eigenvalues are zeros of a holomorphic function .

Abstract

In this paper we study for a given azimuthal quantum number $\kappa$ the eigenvalues of the Chandrasekhar-Page angular equation with respect to the parameters $\mu:=am$ and $\nu:=a\omega$, where $a$ is the angular momentum per unit mass of a black hole, $m$ is the rest mass of the Dirac particle and $\omega$ is the energy of the particle (as measured at infinity). For this purpose, a self-adjoint holomorphic operator family $A(\kappa;\mu,\nu)$ associated to this eigenvalue problem is considered. At first we prove that for fixed $|\kappa|\geq{1/2}$ the spectrum of $A(\kappa;\mu,\nu)$ is discrete and that its eigenvalues depend analytically on $(\mu,\nu)\in\C^2$. Moreover, it will be shown that the eigenvalues satisfy a first order partial differential equation with respect to $\mu$ and $\nu$, whose characteristic equations can be reduced to a Painleve III equation. In addition, we derive a power series expansion for the eigenvalues in terms of $\nu-\mu$ and $\nu+\mu$, and we give a recurrence relation for their coefficients. Further, it will be proved that for fixed $(\mu,\nu)\in\C^2$ the eigenvalues of $A(\kappa;\mu,\nu)$ are the zeros of a holomorphic function $\Theta$ which is defined by a relatively simple limit formula. Finally, we discuss the problem if there exists a closed expression for the eigenvalues of the Chandrasekhar-Page angular equation.