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

TL;DR

This paper investigates generalized solutions to Schrödinger equations with point-like perturbations, comparing different scaling regimes and establishing compatibility with classical solutions within the Colombeau algebra framework.

Contribution

It introduces a unified approach to approximate singular perturbations in Schrödinger equations and analyzes their behavior across various scaling limits, including Colombeau and non-Colombeau regimes.

Findings

Distinct scaling regimes lead to different effective equations.

Compatibility of Colombeau solutions with classical Hartree solutions is demonstrated.

Generalized solutions can be constructed for singular point perturbations.

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.