We show that a polynomial ${\hat{\mathcal{H}}}_{(N)}$ of degree N of a harmonic oscillator hamiltonian allows us to devise a fully solvable continuous quantum system for which the first N discrete energy eigenvalues can be chosen at will. In general such a choice leads to a re-ordering of the associated energy eigenfunctions of $\hat{\mathcal{H}}$ such that the number of their nodes does not increase monotonically with increasing level number. Systems $\hat{\mathcal{H}}$ have certain ‘universal’ features, we study their basic behaviours.

中文翻译：

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具有能量点谱的量子系统的等谱映射到多项式量子谐波振荡器

我们证明了一个多项式 ${\hat{\mathcal{H}}}_{（ñ）}$度Ñ谐波振荡器哈密顿的允许我们设计的量，第一完全解连续量子系统Ñ离散能量本征值可以随意选择。通常，这种选择导致对相关的能量本征函数的重新排序。$\hat{\mathcal{H}}$这样，其节点数不会随级别数的增加而单调增加。系统篇$\hat{\mathcal{H}}$ 具有某些“通用”特征，我们研究它们的基本行为。