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An exactly solvable toy model

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.