We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space $\mathbb {R}^3$ , in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighborhood of the origin.

中文翻译：

---

半线上奇异薛定谔算子的刻画

我们研究了半线上拉普拉斯算子的一​​类类似三角洲的扰动，其特征是原点的罗宾边界条件。使用非标准分析的形式主义，我们推导出与具有非常大（无穷大）幅度和非常短（无穷小）范围的势的薛定谔算子家族的简单联系。因此，在径向势的情况下，我们还得出了欧几里德空间 $\mathbb {R}^3$ 中点相互作用的类似结果 。此外，我们明确讨论了在原点邻域内线性势的情况下的结果。