Schr\"odinger and Klein-Gordon oscillators in Eddington-inspired Born-Infeld gravity: Degree-one Confluent Heun polynomial correspondence

TL;DR

This paper derives conditionally exact solutions for Schr"odinger and Klein-Gordon oscillators in EiBI gravity with magnetic monopoles, revealing how geometric and physical parameters quantize the spectrum through polynomial truncation.

Contribution

It provides the first unified analytical framework for Schr"odinger and Klein-Gordon oscillators in EiBI gravity using degree-one confluent Heun polynomials, linking spectrum quantization to geometric parameters.

Findings

Exact polynomial solutions impose quantization conditions on frequency and angular momentum.

EiBI nonlinearities and monopole deficits determine the admissible bound states.

Relativistic spectrum shows charge symmetry and dependence on physical parameters.

Abstract

We investigate Schr\"odinger 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 Schr\"odinger and Klein-Gordon oscillators in EiBI gravity based on a degree-one confluent Heun polynomial.