For a semi-linear Schrödinger equation of Hartree type in three spatial dimensions, various approximations of singular, point-like perturbations are considered, in the form of potentials of very small range and very large magnitude, obeying different scaling limits. The corresponding nets of approximate solutions represent actual generalised solutions for the singular-perturbed Schrödinger equation. The behaviour of such nets is investigated, comparing the distinct scaling regimes that yield, respectively, the Hartree equation with point interaction Hamiltonian vs the ordinary Hartree equation with the free Laplacian. In the second case, the distinguished regime admitting a generalised solution in the Colombeau algebra is studied, and for such a solution compatibility with the classical Hartree equation is established, in the sense of the Colombeau generalised solution theory.

中文翻译：

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具有点缺陷的线性和非线性薛定谔型方程的广义解：Colombeau 和非 Colombeau 体系

对于三个空间维度的 Hartree 型半线性薛定谔方程，考虑奇异点状扰动的各种近似，其形式为非常小范围和非常大的幅度的势的形式，服从不同的尺度限制。相应的近似解网络代表奇异摄动薛定谔方程的实际广义解。研究了此类网络的行为，比较了分别产生具有点相互作用哈密顿量的哈特里方程与具有自由拉普拉斯算子的普通哈特里方程的不同缩放机制。在第二种情况下，研究了 Colombeau 代数中承认广义解的杰出体系，并在 Colombeau 广义解理论的意义上建立了与经典 Hartree 方程的兼容性。