Abstract

We construct an exactly solvable model of a linear harmonic oscillator with a coordinate-dependent mass in a uniform gravitational field. This model is placed in an infinitely deep potential well with the width \(2a\) and corresponds to the exact solution of the angular part of the Schrödinger equation with one of the Hautot potentials. The wave functions of the oscillator model are expressed in terms of Jacobi polynomials. In the limit \(a\to\infty\), the equation of motion, wave functions, and energy spectrum of the model correctly reduce to the corresponding results of the ordinary nonrelativistic harmonic oscillator with a constant mass. We obtain a new asymptotic relation between the Jacobi and Hermite polynomials and prove it by two different methods.

中文翻译：

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Hautot 势的薛定谔方程的角部分作为在均匀引力场中具有与坐标相关的质量的谐振子

摘要

我们构建了一个完全可解的线性谐振子模型，该模型在均匀引力场中具有与坐标相关的质量。该模型放置在宽度为\(2a\)的无限深势阱中，对应于薛定谔方程的角部分与 Hautot 势之一的精确解。振荡器模型的波函数用雅可比多项式表示。在极限\(a\to\infty\) 内，模型的运动方程、波函数和能谱正确还原为普通非相对论恒质量谐振子的相应结果。我们获得了 Jacobi 多项式和 Hermite 多项式之间的新渐近关系，并通过两种不同的方法证明了它。