论文标题
具有点缺陷的线性和非线性Schrödinger-type方程的通用解决方案：哥伦布和非哥伦布政权
Generalised solutions to linear and non-linear Schrödinger-type equations with point defect: Colombeau and non-Colombeau regimes
论文作者
论文摘要
对于在三个空间维度中的Hartree类型的半线性Schrödinger方程，考虑了各种近似，点状扰动的近似值，以非常小的范围和非常大的电位的形式，遵守不同的尺度限制。近似解决方案的相应网表示奇异扰动的schrödinger方程的实际广义解。研究了此类网的行为，比较了分别比较hartree方程与点相互作用汉密尔顿与普通的哈特里方程与自由laplacian的独特缩放制度。在第二种情况下，研究了在哥伦布代数中承认广义解决方案的杰出政权，并在哥伦布广场广义解决方案理论的意义上建立了与经典哈特里方程的这种解决方案的兼容性。
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.
