We study the Klein-Gordon oscillator in a D-dimensional fractional space with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$1<D\le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>D</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, employing scalar and vector potentials of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S(r)=S_{0}r^{\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>S</mi> <mn>0</mn> </msub> <msup> <mi>r</mi> <mi>λ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$U(r)=U_{0}r^{\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <msup> <mi>r</mi> <mi>λ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, and with power-law coupling <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f(r)=r^{\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>r</mi> <mi>β</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. Using separation of variables in polar coordinates, the angular equation is solved by Gegenbauer polynomials. The radial equation is analyzed for several combinations of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. In different cases, the solutions are expressed in terms of biconfluent Heun functions, Bessel functions, or generalized Laguerre polynomials. The energy spectra and wave functions depend explicitly on the fractional dimension D and on the potential parameters <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda , U_{0}, S_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>,</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>S</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. For specific choices, e.g., <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\beta =\lambda =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>=</mo> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\beta =1, \lambda =-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>λ</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we obtain closed-form energy eigenvalues. For <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\beta =-1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> with a hard-wall confinement, the spectrum is quantized via Bessel zeros. Numerical solutions for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$D=1.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$D=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> show that non-integer dimensions lift degeneracies and modify the number of bound states. These results highlight the sensitivity of relativistic quantum systems to the effective spatial dimensionality and to the power-law exponents of the potentials.

Klein-Gordon Oscillator in D-dimensional Fractional Space

期刊论文
发表日期：

2026年7月14日

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Nader Rouabhia,
Sara Merad,
Mahmoud Merad,
Tolga Birkandan

摘要

We study the Klein-Gordon oscillator in a D-dimensional fractional space with $1<D\le 2$ $1 < D \leq 2$ , employing scalar and vector potentials of the form $S(r)=S_{0}r^{\lambda }$ $S (r) = S_{0} r^{λ}$ and $U(r)=U_{0}r^{\lambda }$ $U (r) = U_{0} r^{λ}$ , and with power-law coupling $f(r)=r^{\beta }$ $f (r) = r^{β}$ . Using separation of variables in polar coordinates, the angular equation is solved by Gegenbauer polynomials. The radial equation is analyzed for several combinations of $\beta $ $β$ and $\lambda $ $λ$ . In different cases, the solutions are expressed in terms of biconfluent Heun functions, Bessel functions, or generalized Laguerre polynomials. The energy spectra and wave functions depend explicitly on the fractional dimension D and on the potential parameters $\lambda , U_{0}, S_{0}$ $λ, U_{0}, S_{0}$ . For specific choices, e.g., $\beta =\lambda =1$ $β = λ = 1$ or $\beta =1, \lambda =-1$ $β = 1, λ = - 1$ , we obtain closed-form energy eigenvalues. For $\beta =-1,$ $β = - 1,$ with a hard-wall confinement, the spectrum is quantized via Bessel zeros. Numerical solutions for $D=1.5$ $D = 1.5$ and $D=2$ $D = 2$ show that non-integer dimensions lift degeneracies and modify the number of bound states. These results highlight the sensitivity of relativistic quantum systems to the effective spatial dimensionality and to the power-law exponents of the potentials.