Schrödinger and Klein-Gordon oscillators in Eddington-inspired Born-Infeld gravity: Degree-one Confluent Heun polynomial correspondence

TL;DR

The study investigates Schrödinger and Klein-Gordon oscillators in a global monopole spacetime within Eddington-inspired Born-Infeld gravity, including Wu-Yang monopole coupling for the relativistic sector. By mapping the radial equations to the confluent Heun form and enforcing polynomial truncation, the authors obtain conditionally exact solutions where the radial functions terminate at a degree $n+1$ and the spectra are fixed by algebraic truncation relations that quantize the oscillator frequency and constrain angular momentum. In the Schrödinger case, the lowest nontrivial state yields $E_n=2\omega\tilde{\alpha}^2\left(n+\frac{5}{2}\right)$ with a closed form for $\omega_0$ and a maximum allowed $\ell$, while in the KG case the energies take the symmetric form $E_0=\pm\sqrt{10\tilde{\alpha}\Omega_0+m_\circ^2}$ with a closed expression for $\Omega_0$ dependent on the WYMM strength, the EiBI parameter, and the angular momentum. The work provides a unified, analytic Heun-polynomial framework for conditional solvability of oscillators in nonlinear gravity with topological defects and magnetic monopoles, revealing how geometric parameters shape the spectrum and offering a reproducible method for extracting analytical results from otherwise non-polynomial Heun equations.

Abstract

We investigate Schrödinger and Klein-Gordon (KG) oscillators in the spacetime of a global monopole (GM) within Eddington inspired Born-Infeld (EiBI) gravity, including, in the relativistic sector, the coupling to a Wu-Yang magnetic monopole (WYMM). By reducing the radial equations to the confluent Heun form and enforcing termination of the Heun series, we obtain conditionally exact solutions in which the radial eigenfunctions truncate to polynomials of degree $(n+1)\geq 1$. This truncation imposes algebraic constraints that quantize the oscillator frequency and restrict the values allowed for the orbital angular momenta $\ell$. In the lowest nontrivial case $n=0$, the degree-one Heun polynomial yields a closed analytic expression for the frequency and determines a finite upper bound on $\ell$, dictated jointly by the EiBI deformation and the GM deficit. The resulting parametric correlations reveal a sharp geometric control of the spectrum: EiBI nonlinearities and the angular deficit fix the admissible bound states through polynomial truncation conditions. The confluent Heun correspondence is made explicit, providing a rigorous and reproducible framework for extracting analytical solutions from otherwise non-polynomial Heun structures. Applying the same method to the KG oscillator with a WYMM, we derive conditionally exact particle and antiparticle energies in a closed form. The relativistic spectrum exhibits perfect charge symmetry and a precise dependence on the WYMM strength, the EiBI parameter and the angular momentum constraint. To the best of our knowledge, this constitutes the first unified and fully consistent treatment of conditionally exact Schrödinger and Klein-Gordon oscillators in EiBI gravity based on a degree-one confluent Heun polynomial.

Figures (2)

• Figure 1: $n=0$ energy states $E_{0,\ell}$ for Schrödinger oscillator in EiBI GM spacetime, computed from Eq. (\ref{['2.15']}). Panels (a)--(c) show the dependence on the gravitational parameter $\tilde{\kappa}$ for fixed effective coupling $\tilde{\alpha}$; panels (d)--(f) show the dependence on $\tilde{\alpha}$ for fixed $\tilde{\kappa}$.
• Figure 2: Energy levels of the KG oscillator in EiBI gravity with the WYMM extension. Each panel displays both positive and negative relativistic branches of the energy eigenvalues obtained from the analytic expression \ref{['3.5']}. (a) Energy levels $E(\tilde{\kappa})$ for the ground state $(n,\ell)=(0,0)$ at $\sigma=0$ and several values of the deformation parameter $\tilde{\alpha}$. (b) $E(\tilde{\kappa})$ for $(n,\ell)=(0,1)$ at $\sigma=0$ and different $\tilde{\alpha}$ values, showing the influence of the angular momentum barrier. (c) $E(\tilde{\kappa})$ for $(n,\ell)=(0,2)$ with fixed $\tilde{\alpha}=0.9$ and increasing WYMM parameter $\sigma$, showing strong sensitivity of the spectrum to $\sigma$. (d) Energy levels as functions of $\tilde{\alpha}$ at fixed $\tilde{\kappa}=0.5$ and $\sigma=3.5$ for $\ell=0,1,2,3$.