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arXiv · 2209.07207

Generalised solutions to linear and non-linear Schr\"{o}dinger-type equations with point defect: Colombeau and non-Colombeau regimes

N. Dugandžija·A. Michelangeli·I. Vojnović
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Abstract

For a semi-linear Schr\"{o}dinger 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 Schr\"{o}dinger 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.

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N. Dugandžija, A. Michelangeli, I. Vojnović. 2022-09-15. Generalised solutions to linear and non-linear Schr\"{o}dinger-type equations with point defect: Colombeau and non-Colombeau regimes. https://arxiv.org/abs/2209.07207

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