标题： 海森堡-维耳代数和李超代数$\mathfrak{osp}\left( {1|2} \right)$的变形：位置相关质量的量子简谐振子的精确解 显示英文标题

标题： Deformation of the Heisenberg-Weyl algebra and the Lie superalgebra $\mathfrak{osp}\left( {1|2} \right)$: exact solution for the quantum harmonic oscillator with a position-dependent mass

摘要： 我们提出了一种新的量子简谐振子海森堡-外尔代数的变形，其中包含一个参数$a>-1$。该参数通过将动量算符$\hat p_x$的定义中的均匀质量$m_0$以及产生-湮灭算符$\hat a^\pm$替换为随位置变化的质量$x$来引入。这种变形的实现是通过在规范方法下对非相对论量子简谐振子对应的薛定谔方程进行精确求解来展示的。所得的能量谱解析表达式由无限多个等距能级组成，而所构建问题的定态波函数则通过厄米特多项式表示。随后，海森堡-外尔代数的变形被推广到李超代数$\mathfrak{osp}\left( {1|2} \right)$的情况。结果表明，这种广义超代数的实现可以应用于抛玻色量子简谐振子问题，其质量表现出与规范变形简谐振子问题中位置依赖质量完全重叠的行为。该问题对于偶数和奇数定态都进行了精确求解。结果表明，变形抛玻色振子的能量谱仍然是等距的，但此时偶数和奇数态的波函数现在都通过拉盖尔多项式表示。还简要讨论了一些恢复常质量规范简谐振子的基本极限关系。 显示英文摘要

摘要： We propose a new deformation of the quantum harmonic oscillator Heisenberg-Weyl algebra with a parameter $a>-1$. This parameter is introduced through the replacement of the homogeneous mass $m_0$ in the definition of the momentum operator $\hat p_x$ as well as in the creation-annihilation operators $\hat a^\pm$ with a mass varying with position $x$. The realization of such a deformation is shown through the exact solution of the corresponding Schr\"odinger equation for the non-relativistic quantum harmonic oscillator within the canonical approach. The obtained analytical expression of the energy spectrum consists of an infinite number of equidistant levels, whereas the wavefunctions of the stationary states of the problem under construction are expressed through the Hermite polynomials. Then, the Heisenberg-Weyl algebra deformation is generalized to the case of the Lie superalgebra $\mathfrak{osp}\left( {1|2} \right)$. It is shown that the realization of such a generalized superalgebra can be performed for the parabose quantum harmonic oscillator problem, the mass of which possesses a behavior completely overlapping with the position-dependent mass of the canonically deformed harmonic oscillator problem. This problem is solved exactly for both even and odd stationary states. It is shown that the energy spectrum of the deformed parabose oscillator is still equidistant, however, both even and odd state wavefunctions are now expressed through the Laguerre polynomials. Some basic limit relations recovering the canonical harmonic oscillator with constant mass are also discussed briefly.