In an attempt to regularize a previously known exactly solvable model [Yang and Zhang, Eur. J. Phys. \textbf{40}, 035401 (2019)], we find yet another exactly solvable toy model. The interesting point is that while the Hamiltonian of the model is parameterized by a function $f(x)$ defined on $[0, \infty )$, its spectrum depends only on the end values of $f$, i.e., $f(0)$ and $f(\infty )$. This model can serve as a good exercise in quantum mechanics at the undergraduate level.

中文翻译：

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一个完全可解的玩具模型

试图对先前已知的完全可解模型进行正则化 [Yang 和 Zhang，Eur。J.物理。\textbf{40}, 035401 (2019)]，我们发现了另一个完全可解的玩具模型。有趣的一点是，虽然模型的哈密顿量由定义在 $[0, \infty )$ 上的函数 $f(x)$ 参数化，它的频谱仅取决于 $f$ 的最终值，即 $f (0)$ 和 $f(\infty )$。这个模型可以作为本科阶段量子力学的一个很好的练习。