Previous works indicate that the frequency ratio of second and firstharmonics of kink oscillations has tendency towards 3 in the case of prominencethreads. We aim to study the magnetohydrodynamic oscillations of longitudinallyinhomogeneous prominence threads and to shed light on the problem of frequencyratio. Classical Sturm--Liouville problem is used for the threads withlongitudinally inhomogeneous plasma density. We show that the spatial variationof total pressure perturbations along the thread is governed by the stationarySchr\"{o}dinger equation, where the longitudinal inhomogeneity of plasmadensity stands for the potential energy. Consequently, the equation has boundedsolutions in terms of Hermite polynomials. Boundary conditions at the threadsurface lead to transcendental dispersion equation with Bessel functions. Thinflux tube approximation of the dispersion equation shows that the frequency ofkink waves is proportional to the expression \alpha(2n+1), where \alpha is thedensity inhomogeneity parameter and n is the longitudinal mode number.Consequently, the ratio of the frequencies of second and first harmonics tendsto 3 in prominence threads. Numerical solution of the dispersion equation showsthat the ratio only slightly decreases for thicker tubes in the case of smallerlongitudinal inhomogeneity of external density, therefore the thin flux tubelimit is a good approximation for prominence oscillations. However, strongerlongitudinal inhomogeneity of external density may lead to the significantshift of frequency ratio for wider tubes and therefore the thin tubeapproximation may fail. The tendency of frequency ratio of second and firstharmonics towards 3 in prominence threads is explained by the analogy of theoscillations with quantum harmonic oscillator, where the density inhomogeneityof the threads plays a role of potential energy.
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