In this note we continue our analysis of the behaviour of self-adjoint Hamiltonians with a pair of identical point interactions symmetrically situated around the origin perturbing various types of "free Hamiltonians" as the distance between the two centres shrinks to zero. In particular, by making the coupling constant to be renormalised dependent also on the separation distance between the centres of the two point interactions, we prove that also in two dimensions it is possible to define the unique self-adjoint Hamiltonian that, differently from the one studied in detail in Albeverio's monograph on point interactions, behaves smoothly as the separation distance vanishes. In fact, we rigorously prove that such a twodimensional Hamiltonian converges in the norm resolvent sense to the one of the negative two-dimensional Laplacian perturbed by a single attractive point interaction situated at the origin having double strength, thus making this two-dimensional model similar to its one-dimensional analogue (not requiring the renormalisation procedure).

中文翻译：

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关于二维哈密顿量的行为 -Δ + λ[δ(~x + ~x0) + δ(~x−~x0)] 随着两个中心之间的距离消失

在这篇笔记中，我们继续分析自伴随哈密顿量的行为，其中一对相同的点相互作用对称地位于原点周围，当两个中心之间的距离缩小到零时，会扰乱各种类型的“自由哈密顿量”。特别是，通过使耦合常数重整化也依赖于两点相互作用中心之间的间隔距离，我们证明了在二维中也可以定义独特的自伴随哈密顿量，它不同于一个在 Albeverio 的关于点相互作用的专着中详细研究，当分离距离消失时表现平稳。实际上，