On the discrete Dirac spectrum of a point electron in the zero-gravity Kerr-Newman spacetime

TL;DR

This paper analyzes the discrete Dirac spectrum for a charged spin-½ particle in the zero-gravity Kerr–Newman spacetime (z$G$KN), proving under explicit smallness conditions that infinitely many eigenvalues lie in the spectral gap and labeling them by a pair of winding numbers arising from coupled cylinder flows. The authors develop a constructive proof based on the Prüfer transform, transforming the Dirac equations into two autonomous 2D dynamical systems on finite cylinders, and show bound states exist precisely when saddles-connectors occur for both theΘ- and Ω-systems; they then couple these analyses via a contraction-mapping argument to obtain joint energy $(E)$ and angular-label $(\lambda)$ values. A hydrogenic dictionary emerges in the $a\to0$ limit, recovering a doubled Sommerfeld spectrum and mapping windings to conventional quantum numbers; numerical experiments reveal rich, oscillatory spectral behavior and hyperfine-like structure induced by the ring geometry. The results provide a rigorous, geometrically flavored framework for understanding how relativistic electrodynamics in a topologically nontrivial background shapes the Dirac spectrum and offer a computational path to explore hydrogenic-like states in this flat but nontrivial spacetime. Overall, the work advances both the mathematical theory of Dirac operators in singular spacetimes and the physical intuition for how ring singularities and anomalous magnetic moments influence spectral features beyond standard quantum mechanics.

Abstract

The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr--Newman spacetime is determined in the zero-$G$ limit (z$G$KN), under some restrictions on the electrical coupling constant and on the radius of the ring-singularity of the z$G$KN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.

Figures (13)

• Figure 1: The universal cover $\widetilde{\mathcal{C}}$ of the finite cylinder $\mathcal{C}$. The fundamental domain $\mathcal{C}_*$ and the equilibrium points are $N^{\pm}_\mu$ and $S^{\pm}_\mu$ are shown. By assumption (b), $N^-_\mu$ is a source, $N^+_\mu$ is a sink, and $S^{\pm}_\mu$ are saddle points.
• Figure 2: The dynamical system for three different parameter values $\mu \in \{\mu_0, \mu_1, \mu_2\}$ are shown. Note that the equilibrium points $S^\pm_\mu$ and $N^\pm_\mu$ vary as $\mu$ varies, but they always remain in the fixed fundamental domain. The winding numbers for $\mu_0$, $\mu_1$, $\mu_2$ are $0$, $1$, $2$, respectively. Note that $\mathcal{W}^-_{\mu_2}$ is a saddles connector while $\mathcal{W}^-_{\mu_0}$ and $\mathcal{W}^-_{\mu_1}$ are not.
• Figure 3: Proof of Proposition \ref{['saddles conn prop']}. At $\mu'$ the winding number is $w_{\mu'} = N$ (the figure on the left shows the case $N = 1$). At $\mu"$ the winding number is $w_{\mu"} = N + 1$ (the figure on the right shows the case $N + 1 = 2$). The signed area of the corridor at $\mu'$ and $\mu"$ are $a(\mu') < 0$ and $a(\mu") > 0$. The intermediate value theorem implies that there exists a $\mu \in [\mu', \mu")$ such that $a(\mu) = 0$ at which we have a saddles connector with winding number $w_\mu = N$.
• Figure 4: The nullclines $\Gamma$ depicted in the fundamental domain $\mathcal{C}_* = [x_-, x_+] \times [y_0, y_0 + 2\pi)$. In this picture we have $P \in \mathcal{A}$ and $Q,R \in \mathcal{B}$.
• Figure 5: The figure on the left represents the barrier for Proposition \ref{['pos wind num theta prop 1']}, while the figure on the right represents the barrier for Proposition \ref{['pos wind num theta prop 2']}. Both figures are for $\kappa > 0$.

Theorems & Definitions (30)

• Theorem 2.1
• Proposition 3.1
• Definition 3.2
• Lemma 3.3
• Proposition 3.4
• Theorem 3.5
• Proposition 3.6
• Lemma 3.7
• Proposition 3.8
• Proposition 3.9