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

Abstract: We solve exactly the relativistic finite-difference equation for the quantum three-dimensional ring-shaped Quesne oscillator potential. Our investigation is based on a finite-difference version of relativistic quantum mechanics. So-called relativistic configurational r-space is a key concept here. We show that the radial wavefunctions and angular wavefunctions are expressed through the continuous dual Hahn polynomials and Jacobi polynomials, respectively. A discrete energy spectrum has been found. The radial wave functions and energy spectrum have the correct nonrelativistic limit. We also build a dynamical symmetry group SU (1, 1) for the radial part of the equation of motion, which allows us to find the energy spectrum purely algebraically.

• The paper presents an exact analytic solution for the relativistic finite-difference equation with the ring-shaped Quesne oscillator potential, yielding discrete bound-state energies.
• The analysis employs orthogonal polynomials and SU(1,1) symmetry to derive quantization conditions and confirm the physical viability of the model.
• The methodology recovers the nonrelativistic limit and offers a foundation for extensions in relativistic quantum mechanics with anisotropic potentials.

Bound State Analysis of the Relativistic Finite-Difference Equation with the Ring-shaped Quesne Oscillator Potential

Introduction and Theoretical Framework

The paper rigorously addresses the exact analytic solution of the relativistic finite-difference equation for a particle subjected to the three-dimensional ring-shaped Quesne oscillator potential. The analysis leverages the finite-difference formulation of relativistic quantum mechanics, wherein the configuration space ($\mathbf{r}$-space) and momentum space are carefully treated to retain manifest covariance and accommodate finite-difference operators. The ring-shaped Quesne oscillator, a generalization of the conventional harmonic oscillator with non-central (ring-shaped) angular dependence, is particularly relevant for modeling molecular and nuclear systems with significant anisotropies.

The finite-difference relativistic Hamiltonian incorporates both radial and angular operators, yielding a nontrivial effective potential landscape that dictates the bound-state spectrum and wavefunction structure. The separation of variables is executed systematically, reducing the problem to coupled angular and radial equations with distinct, analytically tractable forms.

Angular and Radial Solutions: Orthogonal Polynomials and Spectral Quantization

The angular component, after transformation to $x = \cos\vartheta$, leads to a differential equation whose solutions are mapped onto Jacobi and specifically Gegenbauer polynomials. The quantization condition arises naturally from polynomial truncation, fixing the separation constant $\Lambda$ in terms of quantum numbers $k$ and $m$ as $\Lambda = L(L+1)$ with $L = k + |M|$, where $M^2 = m^2 + (2m_0\alpha)/\hbar^2$. The angular wavefunctions manifest as

$$(\sin\vartheta)^{|M|} C_k^{(|M|+1/2)}(\cos\vartheta),$$

preserving orthogonality and normalization over the sphere.

In the radial sector, the analysis exploits the manifestly Hermitian structure of the finite-difference operator. The general solution is expressed in terms of continuous dual Hahn polynomials:

$$R_{nk}(\rho) = \tilde{c}_{nk}(-\rho)^{(\mu_k)}M_{\nu_k}(\rho)S_n(\rho^2; \mu_k, \nu_k, 1/2),$$

where the parameters $x = \cos\vartheta$0, $x = \cos\vartheta$1 encode the details of the effective potential, and $x = \cos\vartheta$2 labels the radial excitation level. The quantization condition for the energy eigenvalues emerges as

$x = \cos\vartheta$3

imposing discrete bound-state spectra.

The approach guarantees that both energy levels and eigenfunctions possess proper nonrelativistic limits, reducing to Laguerre and associated Legendre polynomial solutions in the $x = \cos\vartheta$4 limit, recovering the structural properties of the nonrelativistic Quesne oscillator.

Group-Theoretical Construction: $x = \cos\vartheta$5 Algebra

A significant advancement presented is the demonstration of an $x = \cos\vartheta$6 dynamical symmetry governing the radial subspace. Explicit forms of ladder operators ($x = \cos\vartheta$7) and the Cartan generator ($x = \cos\vartheta$8) are constructed, providing an algebraic route to the energy spectrum, in parallel with the analytic solution. The Casimir operator is evaluated on the basis of bound-state solutions, confirming the irreducibility of the representations encapsulating the radial spectrum. The explicit action of creation and annihilation operators on the wavefunctions substantiates the closure of the algebra and the structure of the Hilbert space.

Numerical and Structural Results: Potential Analysis and Probability Densities

The real and imaginary parts of the effective potential are dissected as functions of the radial coordinate and quantum numbers. The real part exhibits well-defined minima, indicating robust confinement and supporting the existence of discrete bound states; increases in angular momentum shift these minima outward and upward energetically.

Figure 1: Real and imaginary parts of the relativistic ring-shaped Quesne oscillator effective potential $x = \cos\vartheta$9 for different $\Lambda$0.

Analysis of the radial probability density $\Lambda$1 shows systematic extension and emergence of additional nodes/patterns as the radial quantum number $\Lambda$2 increases, indicative of higher excited states.

Figure 2: (a) Radial probability density $\Lambda$3 for several $\Lambda$4 at fixed $\Lambda$5 and $\Lambda$6. (b) Ground state radial function and its probability density.

The dependence of the energy eigenvalues $\Lambda$7 on the ring-shaped coupling $\Lambda$8 is monotonic, with all states shifting upward as $\Lambda$9 increases; the hierarchy of states with respect to angular and radial quantum numbers is preserved throughout.

Figure 3: Variation of relativistic energy eigenvalues $k$0 versus ring-shaped coupling parameter $k$1 for different $k$2 and $k$3.

Three-dimensional probability densities $k$4 for various angular quantum numbers $k$5 are presented in the $k$6–$k$7 plane, revealing a transition from smooth single-mode distributions to highly structured, multi-modal patterns as $k$8 increases, reflecting the underlying angular momentum and its effect on spatial localization.

Figure 4: Three-dimensional spatial probability density $k$9 in the $m$0–$m$1 plane for $m$2 at $m$3.

Physical and Mathematical Implications

The results confirm that the finite-difference relativistic formulation with a non-central ring-shaped oscillator is analytically solvable, with discrete, stable spectra and wavefunctions displaying inherited symmetries from the nonrelativistic domain. The construction of exact bound-state solutions in terms of orthogonal polynomials is significant for applications in relativistic molecular and nuclear quantum mechanics, especially for systems with pronounced anisotropies.

From the group-theoretical perspective, the presence of an $m$4 dynamical symmetry in the radial sector enables potential extensions to coherent state constructions, algebraic derivations of transition matrix elements, and applications to exactly solvable models in quantum optics and quantum information, where non-compact symmetry algebras play a central role.

Numerically, the spectral and probability density analysis substantiates that relativistic corrections (specifically, nontrivial imaginary components in the effective potential) predominantly affect the near-origin regime but do not destabilize the spectrum. The theoretical machinery developed provides a template for further generalizations to potentials with more complex angular and radial dependencies, including additional spin-orbit coupling or external field effects.

Conclusion

The paper presents a comprehensive, exact analytic treatment of the relativistic finite-difference equation for the ring-shaped Quesne oscillator in three dimensions. The explicit construction of bound-state energy spectra, normalized wavefunctions, and the confirmation of an intrinsic $m$5 symmetry establish this model as a canonical example of a solvable relativistic quantum system with non-central potential. The model retains analytic tractability, correct nonrelativistic limit, and spectral stability under general parameter regimes.

These results are instrumental for further theoretical development of exactly solvable relativistic systems, providing a platform for applications to relativistic quantum chemistry, nuclear structure, and potential extensions to field-theoretic frameworks. Future directions include detailed analyses of scattering states, investigation of perturbative corrections, and exploration of the role of these structures in more complex, interacting quantum systems.