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

This paper presents a unified framework for deriving conditionally exact solutions to Schrödinger and Klein-Gordon oscillators in Eddington-inspired Born-Infeld gravity with global and Wu-Yang magnetic monopoles by reducing their radial equations to confluent Heun forms and enforcing degree-one polynomial truncation to quantize frequencies and constrain angular momentum.

Original authors: Omar Mustafa, Abdullah Guvendi

Published 2026-05-06

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CC BY 4.0

Original authors: Omar Mustafa, Abdullah Guvendi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, stretchy fabric. Usually, we think of this fabric as smooth and flat, like a calm lake. But in this paper, the authors are looking at a very specific, slightly bumpy version of that fabric. They are studying what happens to tiny particles (like electrons or other quantum bits) when they are trapped in a "spring" (a harmonic oscillator) inside this bumpy, curved space.

Here is a breakdown of their work using simple analogies:

1. The Setting: A Bumpy Trampoline

The authors are working in a theory called EiBI gravity. Think of this as a special rulebook for how gravity works that is slightly different from Einstein's famous General Relativity.

The Global Monopole: Imagine a trampoline with a heavy, sharp spike stuck in the middle. This creates a "dent" or a "deficit" in the fabric. The space around it isn't perfectly round; it's missing a little slice, like a pizza with a slice taken out.
The Wu-Yang Magnetic Monopole: In the more advanced part of the study, they add a magnetic "knot" to this setup. Think of it as a tiny, invisible tornado of magnetism spinning in the center of the dent.

2. The Experiment: The Quantum Spring

The authors are asking: "If you put a particle on a spring inside this weird, bumpy, magnetic dent, how does it vibrate?"

The Schrödinger Oscillator: This is the "non-relativistic" version. Think of a slow-moving, heavy marble on a spring. It follows the standard rules of quantum mechanics.
The Klein-Gordon (KG) Oscillator: This is the "relativistic" version. Think of a super-fast, light particle (like a photon or an electron moving near light speed) on a spring. This version also accounts for the fact that particles can have "antiparticles" (like a mirror image of the particle).

3. The Problem: A Messy Equation

Usually, when you try to calculate how these particles vibrate in such a complex space, the math becomes a giant, messy knot. The equations are so complicated that they don't have a neat, closed-form answer (like $x = 2$ ). Instead, they usually result in infinite series—endless lists of numbers that you can't easily write down as a single formula.

4. The Solution: The "Stop-Button" Trick

The authors found a clever way to untie this knot. They realized that if the particle's energy and the strength of the "bump" in space are tuned to a very specific relationship, the messy infinite list of numbers suddenly stops.

The Analogy: Imagine a song that is supposed to go on forever. The authors found a specific musical note (a condition) where the song naturally fades out and ends perfectly after just a few bars.
The Result: Because the song stops early, the math turns into a simple polynomial (a short, clean equation) instead of an infinite mess. This is called a "conditionally exact solution." It means the answer is perfect, but only if the universe's parameters (like the size of the dent or the strength of the magnetic knot) fit a specific recipe.

5. The Big Discovery: The "Speed Limit" for Spin

One of the most interesting findings is about angular momentum (how much the particle is spinning or orbiting).

In normal space, a particle can spin at any level.
In this specific bumpy, EiBI gravity space, the authors found a hard limit.
The Analogy: Imagine a carousel. In a normal park, you can have as many horses as you want. But in this specific park (the EiBI space), the ground is so warped that if you try to add too many horses (too much spin), the carousel breaks.
The math shows that for a given energy level, there is a maximum number of spins the particle can have. If it tries to spin faster than that, the "spring" breaks, and the particle can't exist in that state. This limit is dictated by how "bumpy" the space is (the EiBI parameter) and how big the "dent" is (the monopole deficit).

6. The Magnetic Twist

When they added the magnetic "knot" (Wu-Yang monopole) to the fast-moving particles (KG oscillators):

The energy levels of the particle and its "mirror image" (antiparticle) remained perfectly symmetrical. If the particle has energy $+E$ , the antiparticle has energy $-E$ .
The strength of the magnetic knot acted like a volume knob. Turning it up increased the energy of the vibrations, but it also tightened the "speed limit" on how much the particle could spin.

Summary

In short, this paper is a mathematical tour de force. The authors took a very difficult problem involving particles in a warped, magnetic universe and found a "magic key" (a specific tuning of parameters) that turns the impossible, infinite math into a simple, solvable equation.

They discovered that in this specific type of gravity, the universe imposes a strict cap on how much a particle can spin. It's like the universe saying, "You can vibrate and spin, but only up to this specific limit, or the rules of physics in this corner of the universe won't let you exist." This provides a clear, reproducible way to understand how gravity and magnetism shape the quantum world.

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

Problem Statement
This work investigates the behavior of non-relativistic Schrödinger and relativistic Klein-Gordon (KG) oscillators within the spacetime of a global monopole (GM) governed by Eddington-inspired Born-Infeld (EiBI) gravity. The study extends the relativistic sector to include coupling with a Wu-Yang magnetic monopole (WYMM). The primary challenge addressed is obtaining analytical solutions for these oscillators in a curved background where the radial equations typically reduce to non-polynomial Heun differential equations, which generally lack closed-form solutions. The authors aim to establish a framework for extracting "conditionally exact" solutions—analytical results that exist only when specific constraints are imposed on the system's parameters.

Methodology
The authors employ a systematic approach involving the following steps:

Metric and Coupling: They utilize the EiBI gravity metric describing a global monopole, characterized by the Eddington parameter $\tilde{\kappa}$ and the GM deficit parameter $\tilde{\alpha}$ . The oscillators are introduced via non-minimal coupling ( $\partial_j \to \partial_j \pm F_j$ ), where the coupling function $F_j$ is chosen to mimic the harmonic oscillator potential while respecting the curved spacetime geometry.
Reduction to Heun Form: By applying appropriate variable transformations and factoring out exponential prefactors, the radial wave equations for both the Schrödinger and KG oscillators are transformed into the confluent Heun differential equation.
Series Truncation: The core methodological innovation involves enforcing the termination of the confluent Heun power series. Instead of seeking general solutions, the authors impose conditions such that the series truncates to a polynomial of degree $(n+1) \geq 1$ . This requires satisfying specific algebraic constraints on the recursion relations of the series coefficients.
Degree-One Correspondence: Special emphasis is placed on the lowest non-trivial case ( $n=0$ ), where the solution reduces to a degree-one confluent Heun polynomial. The authors rigorously derive the parametric correlations required for this truncation, correcting previous literature that may have utilized incomplete recursion conditions or incorrect potential forms.

Key Results

Schrödinger Oscillator:
- The truncation of the series yields a conditionally exact energy spectrum: $E_n = 2\omega\tilde{\alpha}^2(n + 5/2)$ .
- For the ground state ( $n=0$ ), the oscillator frequency $\omega_0$ is not arbitrary but is determined by a closed analytic expression involving the angular momentum quantum number $\ell$ , the GM parameter $\tilde{\alpha}$ , and the EiBI parameter $\tilde{\kappa}$ .
- A critical finding is the imposition of an upper bound on the orbital angular momentum quantum number $\ell$ . The allowed values of $\ell$ are restricted by the condition $\ell_{\text{max}} = -1/2 + \sqrt{1/4 + (11/3)\tilde{\alpha}^2}$ . For specific ranges of $\tilde{\alpha}$ , only $\ell=0$ or $\ell=0,1$ are permitted.
- The energy levels increase with the monopole parameter $\tilde{\alpha}$ and decrease with the EiBI deformation parameter $\tilde{\kappa}$ .
Klein-Gordon Oscillator with WYMM:
- Applying the same degree-one polynomial correspondence to the KG oscillator (including the WYMM strength $\sigma$ ), the authors derive closed-form expressions for both particle and antiparticle energies: $E_0 = \pm\sqrt{10\tilde{\alpha}\Omega_0 + m_0^2}$ .
- The spectrum exhibits perfect charge symmetry.
- The frequency $\Omega_0$ is constrained by a complex algebraic relation involving $\tilde{\kappa}$ , $\tilde{\alpha}$ , $\ell$ , and the WYMM strength $\sigma$ .
- Similar to the non-relativistic case, a maximum allowed angular momentum is derived: $\ell_{\text{max}} = -1/2 + \sqrt{1/4 + 6\tilde{\alpha}^2 + \sigma^2}$ . This indicates that the WYMM strength and the geometric deficit jointly dictate the number of admissible bound states.

Significance and Claims
The paper claims to provide the first unified and fully consistent treatment of conditionally exact Schrödinger and KG oscillator solutions in EiBI gravity based specifically on a degree-one confluent Heun polynomial. The authors emphasize that their approach offers a rigorous and reproducible framework for extracting analytical solutions from structures that are otherwise non-polynomial.

Key claims regarding the paper's contribution include:

Correction of Literature: The authors assert that previous studies (specifically citing Ahmed and Bouzenada) utilized incorrect recursion relations or potential forms, leading to invalid results for negative signatures or incorrect energy constraints. This work corrects those errors by strictly enforcing the full set of truncation conditions.
Geometric Control: The results demonstrate a "sharp geometric control" of the spectrum, where EiBI nonlinearities and the angular deficit of the global monopole fix the admissible bound states through polynomial truncation conditions.
Analytical Framework: The explicit establishment of the degree-one confluent Heun polynomial correspondence provides a specific mathematical tool for researchers interested in non-trivial polynomial solutions in curved spacetimes, moving beyond numerical approximations for specific parameter sets.

The paper concludes that these findings enrich the set of solvable quantum models in curved and topologically nontrivial spacetimes, laying a foundation for further theoretical investigations into oscillator systems in EiBI geometries.