KG- oscillators in a spinning cosmic string spacetime and an external magnetic field

TL;DR

The study analyzes Klein-Gordon oscillators in a spinning cosmic string spacetime under an external magnetic field. By transforming the KG equation into a radial Schrödinger-oscillator form, the radial solutions are expressed via confluent hypergeometric functions, and the allowed energies are determined by a carefully structured quadratic in $E$ that depends on the spin parameter $β$, wedge parameter $α$, magnetic field $B$, and oscillator parameters. The work distinguishes particle and antiparticle sectors, showing that the presence of the spinning string and magnetic field can cancel wedge effects for certain particle states while breaking energy symmetry between particles and antiparticles in general; large $β$ values lead to clustering of energy levels. In special limits (e.g., $Ω=0$ or $B=0$), the results recover known Landau-like spectra and Symmetry properties, providing a coherent extension of prior findings to rotating string backgrounds. The Appendix supplies a power-series derivation of the exact spectral condition, reinforcing the robustness of the confluent-hypergeometric approach in this curved spacetime setting.

Abstract

We study the Klein-Gordon (KG) oscillators in a spinning cosmic string spacetime and an external magnetic field. The corresponding KG-equation is shown to admit a solution in the form of the confluent hypergeometric functions/polynomials. Consequently, the corresponding energies are shown to be given in a quadratic equation of a delicate nature that has to be solved in an orderly manner (for it involves the energies for KG-particles/antiparticles, $E=E_{\pm}=\pm\left\vert E\right\vert $ along with the magnetic quantum number $m=m_{\pm }=\pm |m|$). Following a case-by-case strategy allowed us to clearly observe the effects of the spinning cosmic string on the spectroscopic structure of the KG-oscillators. We have observed that, under some parametric settings, whilst the Landau-like energies of the KG-particles ($E=E_{+}$), with $m=m_{+}$, have no explicit dependence on the spinning string parameter or wedge parameter $α$, the KG-antiparticles ($E=E_{-}$), with $m=m_{-}$, have such explicit dependence. Interestingly, the co-existence of a spinning cosmic string and external magnetic field eliminates the effect of the wedge parameter ( a byproduct of the string) for KG-particles ($E=E_{+}$), with $m=m_{+}$, but not for the KG-antiparticles ($E=E_{-}$), with $m=m_{-}$. Such co-existence is observed to break the symmetry of the energies of the KG-particles and antiparticles about $E=0$ for KG-oscillators. However, for KG-particles ($E=E_{+}$), with $m=m_{-}$, and KG-antiparticles ($E=E_{-}$), with $m=m_{+}$, are observed to be unfortunate for being indeterminable. Moreover, for the spinning parameter $β>>1$, clustering of the energy levels is observed eminent to indicate that there is no distinction between energy levels at such values of $β$.

Figures (4)

• Figure 1: The energy levels against the magnetic field strength $B$ (for the magnetic quantum number $m=0$ and $n_{r}=0,1,2,3$) given by Eq. (\ref{['2.10']}), at $\alpha =0.5$, and $m_{\circ }=1=e=k=\Omega$, so that Fig.1(a) for $\beta =0$ ( no spinning of the cosmic string), 1(b) for $\beta =1$, and 1(c) for $\beta =2$.
• Figure 2: The energy levels against the KG-oscillators' frequency $\Omega$ (for $m=0$ and $n_{r}=0,1,2,3$) given by Eq. (\ref{['2.10']}), at $\alpha =0.5$, and $m_{\circ }=1=e=k=B$, so that 2(a) for $\beta =0$, 2(b) for $\beta =1$, and 2(c) for $\beta =2$.
• Figure 3: The energy levels (\ref{['2.19']}) against the magnetic field strength $B$ (for $m=1\neq 0$ and $n_{r}=0,1,2,3$ at $\alpha =0.5$, and $m_{\circ }=1=e=k=\Omega$) so that the particles' energies are given in 3(a) for $\beta =0$, 3(b) for $\beta =1$, and 3(c) for $\beta =2$. The anti-particles' energies are given in 3(d) for $\beta =0$, 3(e) for $\beta =1$, and 3(f) for $\beta =2$.
• Figure 4: The energy levels (\ref{['2.19']}) against the KG-oscillators' frequency $\Omega$ (for $m=1\neq 0$ and $n_{r}=0,1,2,3$ at $\alpha =0.5$, and $m_{\circ }=1=e=k=B$) so that the particles' energies are given in 4(a) for $\beta =0$, 4(b) for $\beta =1$, and 4(c) for $\beta =2$. The anti-particles' energies are given in 4(d) for $\beta =0$, 4(e) for $\beta =1$, and 4(f) for $\beta =2$.