Through the Kre\u{\i}n-Vi\v{s}ik-Birman extension scheme, unlike the classical analysis based on von Neumann's theory, we reproduce the construction and classification of all self-adjoint realisations of three-dimensional hydrogenoid-like Hamiltonians with singular perturbation supported at the Coulomb centre (the nucleus), as well as of Schr\"{o}dinger operators with Coulomb potentials on the half-line. These two problems are technically equivalent, albeit sometimes treated by their own in the the literature. Based on such scheme, we then recover the formula to determine the eigenvalues of each self-adjoint extension, as corrections of the non-relativistic hydrogenoid energy levels. We discuss in which respect the Kre\u{\i}n-Vi\v{s}ik-Birman scheme is somewhat more natural in yielding the typical boundary condition of self-adjointness at the centre of the perturbation.

中文翻译：

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具有中心扰动的类氢光谱

通过 Kre\u{\i}n-Vi\v{s}ik-Birman 扩展方案，不同于基于冯诺依曼理论的经典分析，我们再现了三维氢形体的所有自伴随实现的构建和分类-像在库仑中心（原子核）上支持奇异扰动的哈密顿量，以及在半线上具有库仑势的 Schr\"{o}dinger 算子。这两个问题在技术上是等价的，尽管有时由它们自己处理在文献中。基于这样的方案，我们然后恢复公式来确定每个自伴随扩展的特征值，作为非相对论氢形能级的修正。我们讨论在哪个方面 Kre\u{\i}n-Vi\v{s}ik-Birman 方案在产生扰动中心的自邻接的典型边界条件方面更自然一些。