We formulated the oscillators with position-dependent finite symmetric decreasing and increasing mass. The classical phase portraits of the systems were studied by analytical approach (He’s frequency formalism). We also study the quantum mechanical behaviour of the system and plot the quantum mechanical phase space for necessary comparison with the same obtained classically. The phase portrait in all the cases exhibited closed loop reflecting the stable system but the quantum phase portrait exhibited the inherent signature (cusp or kink) near origin associated with the mass. Although the systems possess periodic motion, the discrete eigenvalues do not possess any similarity with that of the simple harmonic oscillator having m = 1.

中文翻译：

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像振荡器一样的位置相关的有限对称质量谐波：经典和量子力学研究

我们用与位置有关的有限对称减小和增加质量来构造振荡器。通过分析方法（他的频率形式主义）研究了系统的经典相图。我们还研究了系统的量子力学行为，并绘制了量子力学相空间，以便与经典获得的相比较。在所有情况下，相图都表现出反映稳定系统的闭环，但是量子相图在与质量相关的原点附近表现出固有的特征（尖峰或扭结）。尽管系统具有周期性运动，但是离散特征值与具有m = 1的简单谐波振荡器没有任何相似性。