In this article, we answer the following question: If the wave equation possesses bound states but it is exactly solvable for only a single non-zero energy, can we find all bound state solutions (energy spectrum and associated wavefunctions)? To answer this question, we use the "tridiagonal representation approach" to solve the wave equation at the given energy by expanding the wavefunction in a series of energy-dependent square integrable basis functions in configuration space. The expansion coefficients satisfy a three-term recursion relation, which is solved in terms of orthogonal polynomials. Depending on the selected energy we show that one of the potential parameters must assume a value from within a discrete set called the "potential parameter spectrum" (PPS). This discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite discrete set. Inverting the relation between the energy and the PPS gives the bound state energy spectrum. Therefore, the answer to the above question is affirmative.

中文翻译：

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束缚态和势参数谱

在本文中，我们回答以下问题：如果波动方程具有束缚态，但它只能对单个非零能量精确求解，我们能否找到所有束缚态解（能谱和相关波函数）？为了回答这个问题，我们使用“三对角表示法”通过在配置空间中一系列与能量相关的平方可积基函数中展开波函数来求解给定能量下的波动方程。展开系数满足三项递归关系，该关系是根据正交多项式求解的。根据所选的能量，我们表明其中一个势参数必须假定来自称为“势参数谱”(PPS) 的离散集内的值。这个离散集是从上述多项式的频谱中获得的，可以是有限的或无限的离散集。反转能量和 PPS 之间的关系给出束缚态能谱。因此，上述问题的答案是肯定的。