In this paper, exactly solvable model of the quantum harmonic oscillator is proposed. Wave functions of the stationary states and energy spectrum of the model are obtained through the solution of the corresponding Schrödinger equation with the assumption that the mass of the quantum oscillator system varies with position. We have shown that the solution of the Schrödinger equation in terms of the wave functions of the stationary states is expressed by the pseudo Jacobi polynomials and the mass varying with position depends from the positive integer N. As a consequence of the positive integer N, energy spectrum is not only non-equidistant, but also there are only a finite number of energy levels. Under the limit, when N →∞, the dependence of effective mass from the position disappears and the system recovers known non-relativistic quantum harmonic oscillator in the canonical approach where wave functions are expressed by the Hermite polynomials.

中文翻译：

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离散值的有效质量作为一维谐振子模型的隐藏特征：质量随位置变化的薛定谔方程的精确解

本文提出了量子谐振子的精确可解模型。假设量子振子系统的质量随位置变化，通过求解相应的薛定谔方程得到模型的稳态波函数和能谱。我们已经证明，薛定谔方程关于稳态波函数的解由伪雅可比多项式表示，并且随位置变化的质量取决于正整数ñ. 由于正整数ñ，能谱不仅不等距，而且只有有限数量的能级。在极限之下，当ñ →∞，有效质量对位置的依赖性消失了，系统恢复了规范方法中已知的非相对论量子谐振子，其中波函数由 Hermite 多项式表示。