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Bound State Solutions of the Relativistic Finite-difference Equation for the Ring-shaped Quesne Oscillator Potential

This paper presents an exact solution to the relativistic finite-difference equation for the ring-shaped Quesne oscillator potential, deriving discrete energy spectra and wavefunctions expressed via continuous dual Hahn and Jacobi polynomials, while establishing an SU(1,1) dynamical symmetry group and analyzing the system's dependence on the parameter $\alpha$ .

Original authors: Shakir Nagiyev, Narmin Nasibova, Vafa Tarverdiyeva, Gulnara Guliyeva

Published 2026-07-02

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CC BY 4.0

Original authors: Shakir Nagiyev, Narmin Nasibova, Vafa Tarverdiyeva, Gulnara Guliyeva

Original paper licensed under CC BY 4.0 (https://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 you are trying to understand how a tiny particle, like an electron, behaves when it's trapped inside a very specific, weird-shaped cage. In the world of everyday physics (non-relativistic), we have a standard rulebook called the Schrödinger equation to predict where the particle might be. But when that particle moves incredibly fast—close to the speed of light—those old rules break down, and we need a new, more complex rulebook called "relativistic quantum mechanics."

This paper by Nagiyev and colleagues is like a master key that unlocks a specific, complicated version of this high-speed puzzle. Here is a simple breakdown of what they did, using everyday analogies:

1. The "Ring-Shaped" Cage

Usually, we think of particles trapped in a spherical ball or a simple box. But this paper looks at a "Ring-shaped Quesne Oscillator."

The Analogy: Imagine a particle running on a track that isn't just a circle, but a track that gets tighter and tighter as it goes up and down, shaped like a donut with a very specific, mathematical curve.
The Twist: The authors didn't just use the standard rules for fast particles. They used a "finite-difference" approach. Think of standard physics as a smooth, continuous movie. This new approach treats space and time like a pixelated video game. Instead of moving smoothly, the particle "jumps" from one pixel to the next. This method is often better for handling the extreme speeds where the particle's mass and energy get weird.

2. Solving the Puzzle (The Math Magic)

The authors wanted to know two things:

Where is the particle likely to be? (The Wave Function)
How much energy does it have? (The Energy Spectrum)

To find the answers, they had to split the problem into two parts, like solving a Rubik's cube by looking at the top face and the side face separately:

The Side Face (Angular Part): They found that the particle's movement around the ring follows a pattern described by Jacobi polynomials. Think of these as a specific type of musical rhythm that tells you how the particle vibrates as it goes around the ring.
The Top Face (Radial Part): This is the part dealing with how far the particle is from the center. They discovered this follows a pattern called Continuous Dual Hahn polynomials. If the Jacobi polynomials are the rhythm, these are the specific notes the particle plays as it moves in and out.

3. The "Hidden Symmetry" (The Secret Code)

One of the coolest things they found is that this system has a "Dynamical Symmetry Group" called SU(1, 1).

The Analogy: Imagine you have a set of Lego blocks. Usually, you have to build them piece by piece. But if you discover a "magic instruction manual" (the symmetry group), you realize you can build the whole tower just by stacking one specific block over and over again in a predictable way.
The Result: Because they found this "magic manual," they could calculate the energy levels of the particle using pure algebra (math logic) without having to do messy, long calculations every time. It's like knowing the answer to a math problem just by looking at the pattern of the numbers.

4. What Happens When You Turn the Dials?

The authors ran computer simulations to see how the system changes when you tweak the "ring" shape (changing a parameter called $\alpha$ ).

The Observation: As they made the "ring" interaction stronger, the energy levels of the particle didn't jump around randomly. Instead, they monotonically shifted upward.
The Analogy: Imagine a guitar string. If you tighten the string (increase the parameter), the pitch (energy) gets higher and higher. The paper shows that for this specific relativistic ring, tightening the "string" always makes the energy go up, and the particle stays safely trapped in its "donut" cage.

5. The "Real" vs. "Imaginary" Parts

Because they are using this "pixelated" (finite-difference) relativistic math, the potential energy (the force holding the particle) has two parts: a "Real" part and an "Imaginary" part.

The Real Part: This is the main force, like the walls of the cage. It creates a deep valley where the particle likes to sit.
The Imaginary Part: This is a subtle, weird correction that only appears because the particle is moving so fast and the math is "pixelated." It's most noticeable when the particle is very close to the center, but it doesn't change the main story; it just adds a tiny, relativistic "flavor" to the mix.

The Bottom Line

The authors successfully took a complex, high-speed physics problem involving a weird ring-shaped trap and solved it exactly. They proved that:

The math works perfectly and matches our old, slow-speed physics when the particle slows down.
The particle's behavior is governed by beautiful, known mathematical patterns (polynomials).
The system is stable: even as you change the shape of the trap, the energy levels just shift up in a predictable, orderly fashion.

In short, they built a perfect, mathematical model of a fast-moving particle in a ring-shaped cage, showing us exactly how it dances and what energy it needs to keep dancing.

Technical Summary: Bound State Solutions of the Relativistic Finite-difference Equation for the Ring-shaped Quesne Oscillator Potential

Problem Statement
The paper addresses the challenge of finding exact bound state solutions for a quantum particle moving in a three-dimensional ring-shaped Quesne oscillator potential within the framework of relativistic quantum mechanics. While the non-relativistic Schrödinger equation effectively describes low-energy phenomena, high-energy dynamics require relativistic formulations. Although the Klein-Gordon and Dirac equations are standard, the authors focus on the finite-difference version of relativistic quantum mechanics, which utilizes a relativistic configurational $r$ -space and a momentum space realized on the upper sheet of the mass hyperboloid (Lobachevsky space). The specific goal is to generalize the exactly solvable non-relativistic Quesne ring-shaped oscillator potential to this relativistic finite-difference context and to construct the associated dynamical symmetry group.

Methodology
The authors employ a functional approach to solve the relativistic finite-difference wave equation:
$[H_0 + V(r)]\psi(r) = E\psi(r)$
where $H_0$ is the finite-difference free Hamiltonian and $V(r)$ is the ring-shaped Quesne potential. The potential is defined as:
$V(r) = \left[ \frac{1}{2}m_0\omega^2(r + i\lambda)^2 + \frac{\alpha}{r^2 \sin^2 \vartheta} \right] e^{i\lambda \partial_r}$
where $\lambda = \hbar/(m_0c)$ is the Compton wavelength and $\alpha$ is a dimensionless parameter related to the ring-shaped interaction strength.

The solution process involves the following steps:

Separation of Variables: The wave function is separated into radial ( $R(r)$ ), angular ( $F(\vartheta)$ ), and azimuthal ( $e^{im\phi}$ ) components. This reduces the problem to a finite-difference radial equation and a second-order differential angular equation.
Angular Solution: The angular equation is transformed into a form solvable by Gegenbauer polynomials (a specific case of Jacobi polynomials). The separation constant $\Lambda$ is quantized based on the angular quantum numbers $k$ and $m$ .
Radial Solution: The radial equation is solved by identifying its structure with the difference equation satisfied by continuous dual Hahn polynomials. The authors determine the asymptotic behavior of the wave function at $r=0$ and $r=\infty$ to define the polynomial solution.
Dynamical Symmetry Construction: The authors construct a dynamical symmetry group, SU(1, 1), for the radial equation. They define raising and lowering operators ( $K_+$ and $K_-$ ) and a Casimir operator to derive the energy spectrum algebraically, without relying solely on the functional solution.
Non-Relativistic Limit: The derived energy spectrum and wave functions are analyzed in the limit $c \to \infty$ to verify their consistency with known non-relativistic results.

Key Contributions and Results

Exact Analytical Solutions: The paper provides exact analytical expressions for the bound state wave functions and energy spectrum. The radial wave functions are expressed in terms of continuous dual Hahn polynomials, while the angular wave functions are expressed via Jacobi (Gegenbauer) polynomials.
Energy Spectrum Quantization: A discrete energy spectrum is derived:
$E_{nk} = \hbar\omega(2n + \mu_k + \nu_k)$
where $n$ is the radial quantum number, and $\mu_k, \nu_k$ are parameters dependent on the system's physical constants and the separation constant $\Lambda_k$ .
Dynamical Symmetry Group: The radial part of the equation is shown to possess an SU(1, 1) dynamical symmetry. This allows the energy spectrum to be determined purely algebraically using the representation theory of the group.
Non-Relativistic Limit: The study confirms that the relativistic solutions correctly reduce to the standard non-relativistic ring-shaped oscillator solutions (involving Laguerre polynomials) in the limit $c \to \infty$ .
Numerical Analysis: Numerical results demonstrate that:
- The effective potential possesses a real part with a pronounced minimum (confining region) and an imaginary part that acts as a relativistic correction, primarily significant at small radial distances.
- The energy spectrum shifts monotonically toward higher energies as the ring-shaped coupling parameter $\alpha$ increases.
- The probability densities exhibit distinct quantum structures that evolve with the angular quantum number $k$ .

Significance and Claims
The paper claims to provide an exactly solvable framework for investigating relativistic quantum systems with non-central (ring-shaped) interactions. By successfully generalizing the Quesne potential to the finite-difference relativistic formalism, the authors establish a model that retains analytical tractability while incorporating relativistic effects.

The significance lies in:

Theoretical Extension: It extends the class of exactly solvable problems in relativistic quantum mechanics beyond central potentials to include ring-shaped potentials.
Algebraic Structure: The identification of the SU(1, 1) dynamical symmetry group offers a group-theoretical method for solving the radial equation, complementing the functional approach.
Physical Insight: The model serves as a platform for exploring relativistic bound states, hidden symmetries, and anisotropic quantum systems. The authors note that the ring-shaped interaction acts as an additional confinement mechanism, stabilizing the spectrum while shifting energy levels.

The work does not propose new experimental setups but rather offers a theoretical tool for understanding the dynamics of particles in deformed potentials (relevant to nuclear physics and molecular physics) within a rigorous relativistic finite-difference framework.