Analytical Solutions of the Generalized Klein-Gordon Oscillator in Som-Raychaudhuri Space-Time via the Extended Nikiforov-Uvarov Method

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Overview

Research area: Relativistic quantum mechanics in curved space–time — specifically, exactly solvable scalar (spin-0) wave equations in a rotating Gödel-type background (the Som–Raychaudhuri metric), treated with a generalized version of the Nikiforov–Uvarov (NU) differential-equation method.

Technical level: Advanced. The paper assumes familiarity with the Klein–Gordon equation in curved space–time, covariant derivatives and minimal coupling, cylindrical metrics with vorticity and angular deficit, and special functions (biconfluent Heun polynomials). The results are presented as closed-form algebraic conditions plus numerical tables; there is no pedagogical derivation of the underlying formalism.

Scope (one sentence): The authors derive exact energy eigenvalues and radial eigenfunctions for the generalized Klein–Gordon oscillator in Som–Raychaudhuri space–time under a uniform magnetic field, for a linear-type and a Cornell-type oscillator function, using the extended Nikiforov–Uvarov method.

What This Paper Is About

The paper asks how a relativistic quantum oscillator (the Klein–Gordon oscillator) behaves when it sits in a rotating, topologically defective background — Som–Raychaudhuri space–time, which contains an angular-deficit parameter α and a vorticity parameter Ω — while also being subject to a uniform magnetic field and a scalar potential. The core difficulty is that the resulting radial equations are Heun-type, which in the literature are usually solved only case-by-case, with ansatz guesses or by truncating infinite series. The goal is to show that the extended Nikiforov–Uvarov method solves these equations directly and algebraically, yielding explicit quantization conditions and closed-form wave functions for two interaction models: a linear potential and a Cornell potential.

Key Contributions

1. A systematic application of the extended NU method to a relativistic wave equation in curved space–time. The authors frame this as a "unified algebraic solution framework" intended as an alternative to "non-unified and problem-specific methods," and they apply it to the generalized Klein–Gordon oscillator in the Som–Raychaudhuri metric with a uniform magnetic field.

2. Exact quantization conditions and closed-form eigenfunctions for two potentials. For the linear potential (oscillator function f(r) = r with Coulomb-type scalar potential S(r) = λ/r) and the Cornell potential (f(r) = ξ₁r + ξ₂/r, again with S(r) = λ/r), the paper obtains energy conditions purely algebraically, "without the need for series truncation," and writes the radial wave functions in terms of biconfluent Heun polynomials.

3. A branch-selection rule for physically admissible solutions. The extended formalism generates four possible forms of the polynomial π_e(r); the paper identifies which are physically acceptable by requiring τ′_e(r) < 0, and finds that only π_e2(r) and π_e3(r) satisfy this condition, yielding two distinct energy relations (Eqs. 30 and 37 for the linear case; Eqs. 48 and 51 for the Cornell case).

4. Numerical energy spectra and graphical analysis. Two tables report positive energy eigenvalues for the topological parameter values α = 0.3, 0.5 and 0.8, and a four-panel figure shows the radial wave functions at α = 0.8 for the two solution branches of each potential.

Main Findings

• Two admissible quantization conditions per potential. For the linear potential the energy conditions are 2n = a₄/a₂ − 2 + 2√a₁ (from π_e2) and 2n = a₄/a₂ − 2 − 2√a₁ (from π_e3), with C_n = −2a₃ in both cases. The Cornell case mirrors this structure with b-coefficients: 2n = b₄/b₂ − 2 ± 2√b₁.

• Explicit closed-form wave functions. The radial solutions take the form ψ(r) = exp(−a₂r²/2) r^(∓√a₁) N(∓2√a₁, 0, a₄/a₂, 4a₃/√a₂, √a₂ r), where N is a biconfluent Heun polynomial. The corresponding Cornell solutions have the same structure with b₁, b₂, b₃, b₄ replacing a₁, a₂, a₃, a₄.

• Consistency with the biconfluent Heun condition. Comparing the radial equation with the standard biconfluent Heun form gives α = −2√a₁, β = 0, γ = a₄/a₂, δ = 4a₃/√a₂, and the paper shows that the truncation requirement γ − α − 2 = 2n coincides exactly with the eigenvalue relation obtained independently by the extended NU method.

• The topological parameter α affects states with angular momentum differently from l = 0 states. For l = 0 states the energies slightly increase as α increases; for l ≠ 0 states the energies decrease as α increases. The paper attributes this to the effective angular momentum term ˘l_eff/α, which "plays a dominant role in shifting the relativistic energy levels," and states that increasing α "weakens the influence of the topological defect."

• Representative numbers (linear potential, all parameters set to 1). For (n, l) = (0, 0): 4.53166 at α = 0.3, 4.7502 at α = 0.5, 4.90384 at α = 0.8. For (n, l) = (2, 1): 18.2741, 14.9574, 11.8368 at the same α values — the opposite trend. For (n, l) = (5, 4): 63.8521, 43.4488, 32.0359.

• Cornell-type energies exceed linear-type energies for the same quantum numbers. For (n, l) = (0, 0) the Cornell values are 6.14859, 6.3820 and 6.5364 at α = 0.3, 0.5, 0.8, compared with 4.53166, 4.7502 and 4.90384 for the linear case. For (n, l) = (5, 4) the Cornell values are 64.0362, 43.7336 and 32.4486. The paper attributes this to "the stronger confining character of the Cornell-type interaction."

• An Aharonov–Bohm-like shift. The effective angular momentum is ˘l_eff = l − eΦ_B/(2π); the paper states that this modifies the energy spectrum, "leading to an effect analogous to the Aharonov–Bohm effect."

• Agreement and disagreement with prior work. The results are stated to be consistent with Ref. [10], where energies decrease with increasing α for nonzero angular momentum; however, the present analysis finds a different trend for l = 0 states, which the paper highlights as an additional observation.

• Bound-state behaviour. All radial wave functions shown vanish at large distances, supporting their interpretation as bound states; higher quantum states spread over a larger radial region and show more pronounced oscillation. The four panels (a)–(d) also show that the different admissible branches of the extended NU method produce distinct radial profiles from the same wave equation.

Methodology in Plain English

The authors start with the Klein–Gordon equation written in the Som–Raychaudhuri metric, whose line element contains an angular-deficit parameter α (with α = 1 − 4η, where η is a linear mass density) and a vorticity parameter Ω. They minimally couple the electromagnetic field through a four-potential with only an azimuthal component, A_φ = −(αB₀/2)r² + Φ_B/(2π), which encodes both a uniform magnetic field B₀ and a constant internal flux Φ_B, and they add the oscillator coupling through a radial function f(r). They then separate variables with the ansatz Ψ = exp[i(−Et + lφ + kz)] s(r), leaving a single radial equation.

The key step is rewriting that radial equation into a canonical form in which the coefficient functions are polynomials of higher degree than the standard Nikiforov–Uvarov recipe allows — the paper states maximum degrees of two, three and four for τ̃_e, σ_e and σ̃_e respectively. This is what makes the method applicable to equations whose solutions are Heun-type rather than the classical hypergeometric cases. Inside this framework one constructs an auxiliary polynomial π_e(r) from a square-root relation, forms τ_e = τ̃_e + 2π_e, and discards any branch where τ′_e is not negative (those do not give normalizable bound states). The quantization condition is then imposed by matching the function h(r) to its degree-n form h_n(r); this matching produces both the energy condition and the integration constant.

Finally, the energy condition is solved numerically for the positive energy eigenvalues using a common parameter set (M = e = Φ_B = ω = λ = B₀ = Ω = k = 1, plus ξ₁ = 2 and ξ₂ = 1 for the Cornell case) with n = 0, 1, 2, … and l = 0, 1, …, n − 1, and the wave functions — built from an exponential factor, a power r^(±√a₁) and a biconfluent Heun polynomial — are plotted up to an arbitrary normalization constant.

Why This Matters

Impact on research. The paper's contribution is methodological rather than observational: it argues that one algebraic framework can generate both the spectrum and the eigenfunctions for a class of curved-space relativistic oscillator problems that are otherwise attacked with problem-specific ansätze or series truncation. If the claim holds, it standardizes a calculation that currently appears repeatedly in the literature in slightly different forms, and it makes the dependence on topological defect parameters, rotation and magnetic flux explicitly traceable in the spectrum. The authors suggest the framework extends to Heun-type equations generally and to relativistic wave equations beyond this one.

Real-world applications. The paper does not claim any engineering or experimental applications, and it explicitly states that the study "is based entirely on analytical calculations" with no experimental or observational data. The physical contexts it situates itself in are:

• Cosmic strings and other topological defects arising from phase transitions in the early universe, which motivate the angular-deficit parameter α.
• Rotating matter distributions in general relativity, via the Gödel-type and Som–Raychaudhuri metrics, including questions about rotation and potential causality violations.
• Quark modelling, since the Cornell potential is introduced as a potential "which describes behavior of quarks."
• Analogue Landau-level physics: the paper notes that spectra in Gödel-type backgrounds show "notable similarities with the Landau problem in flat, spherical, and hyperbolic geometries," which connects to the broader programme of simulating gravitational backgrounds in tabletop quantum systems.

Industry relevance. Not claimed in the paper. Any industrial relevance would be indirect — through the mathematical toolkit (biconfluent Heun functions, algebraic quantization conditions) used in computational physics software, or through analogue condensed-matter systems — but the paper presents no device, material or technology result.

Future Directions

• Other Gödel-type backgrounds. The paper works in the flat (Som–Raychaudhuri) limit where the curvature parameter vanishes; applying the same extended NU machinery to Gödel-type metrics with nonzero curvature parameters is a natural extension, particularly since the introduction notes that spectra in those backgrounds resemble the Landau problem in flat, spherical and hyperbolic geometries.
• Fermions and torsion. The introduction reviews prior work on Dirac fermions, Weyl fermions, the Dirac oscillator and particles in Som–Raychaudhuri space–time with torsion and cosmic strings, where degeneracies are lifted and eigenfunctions are affected by topological defects; the authors' framework would need extension to spin-1/2 equations in these backgrounds.
• Kaluza–Klein and extra-dimensional reductions. The authors explicitly expect the framework to apply to "Kaluza–Klein reductions and effective Schrödinger-type equations induced by extra-dimensional theories."
• Wider potential classes and the l = 0 discrepancy. The paper reports a trend for l = 0 states that differs from Ref. [10]; resolving that difference, and testing whether the branch-selection rule (τ′_e < 0) remains the decisive criterion for other potentials and field configurations, are open questions.

Target Audience

Graduate students and researchers in theoretical and mathematical physics working on exactly solvable relativistic wave equations, especially those interested in quantum mechanics in curved space–time, Gödel-type and cosmic-string backgrounds, and the Klein–Gordon and Dirac oscillators. It is also relevant to specialists in special functions who work with Heun-type equations and to readers interested in the extended Nikiforov–Uvarov method as a general-purpose solving technique. Readers seeking experimental tests, numerical simulation code, or applied device proposals will not find them here.

Authors’ abstract

In this study, we present an analytical approach based on the extended Nikiforov-Uvarov method to solve the generalized Klein-Gordon oscillator in the presence of a uniform magnetic field within the Som-Raychaudhuri space-time. Exact eigenstate solutions are obtained for two distinct potential models, namely the linear potential and the Cornell potential. The corresponding energy eigenvalues are determined, and the associated eigenfunctions are analyzed and illustrated graphically.