Abstract
A singular sextic oscillator in the finite-Hermite biconfluent-Heun hierarchy generates an exact four-member power-Liouville Hermite orbit. Power coordinate transformations with the Liouville amplitude map the hierarchy member $N=4$ to four half-line Schrödinger spectral charts while preserving the physical regular-recessive Wronskian zero set under energy-coupling reparametrization. Two charts are the previously studied $x^{2/3}$ and inverse-square-root conditionally integrable potentials; the orbit embeds this known power-dual pair together with the singular sextic and linear-plus-quadratic charts. On the selected physical slice, the degree-five accessory compatibility obstruction vanishes identically, while the transformed Frobenius index reproduces both published fractional-power centrifugal coefficients. The common endpoint condition yields an exact discrete sequence of quartic Sturmian eigencouplings for the sextic oscillator. Independent two-sided shooting confirms representative eigencouplings and node counts. A matched large-action analysis further derives the first matched-endpoint correction within the leading Langer action and improves the Maslov-shifted effective-level approximation. The construction thus connects exact Hermite solvability, energy quantization, coupling quantization, and power duality within a single spectral orbit.
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Cite this article
APA 7
Ishkhanyan, A., & Znojil, M. (2026). A Power-Liouville Hermite Orbit: Common Spectral Quantization of the Singular Sextic Oscillator and Fractional-Power Potentials. https://omanscience.com/en/articles/a-power-liouville-hermite-orbit-common-spectral-quantization-of-the-singular-sextic-oscillator-and-fractional-power-potentials
MLA 9
Ishkhanyan, Artur, and Miloslav Znojil. "A Power-Liouville Hermite Orbit: Common Spectral Quantization of the Singular Sextic Oscillator and Fractional-Power Potentials." https://omanscience.com/en/articles/a-power-liouville-hermite-orbit-common-spectral-quantization-of-the-singular-sextic-oscillator-and-fractional-power-potentials.
Chicago (author–date)
Ishkhanyan, Artur, and Miloslav Znojil. 2026. "A Power-Liouville Hermite Orbit: Common Spectral Quantization of the Singular Sextic Oscillator and Fractional-Power Potentials." https://omanscience.com/en/articles/a-power-liouville-hermite-orbit-common-spectral-quantization-of-the-singular-sextic-oscillator-and-fractional-power-potentials.
Harvard
Ishkhanyan, A. and Znojil, M. (2026) 'A Power-Liouville Hermite Orbit: Common Spectral Quantization of the Singular Sextic Oscillator and Fractional-Power Potentials', Available at: https://omanscience.com/en/articles/a-power-liouville-hermite-orbit-common-spectral-quantization-of-the-singular-sextic-oscillator-and-fractional-power-potentials.
Vancouver
Ishkhanyan A, Znojil M. A Power-Liouville Hermite Orbit: Common Spectral Quantization of the Singular Sextic Oscillator and Fractional-Power Potentials. https://omanscience.com/en/articles/a-power-liouville-hermite-orbit-common-spectral-quantization-of-the-singular-sextic-oscillator-and-fractional-power-potentials
IEEE
A. Ishkhanyan, and M. Znojil, "A Power-Liouville Hermite Orbit: Common Spectral Quantization of the Singular Sextic Oscillator and Fractional-Power Potentials," https://omanscience.com/en/articles/a-power-liouville-hermite-orbit-common-spectral-quantization-of-the-singular-sextic-oscillator-and-fractional-power-potentials.