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Generalised solutions to linear and non-linear Schrödinger-type equations with point defect: Colombeau and non-Colombeau regimes

N. Dugandžija, A. Michelangeli, I. Vojnović

Abstract

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.

Generalised solutions to linear and non-linear Schrödinger-type equations with point defect: Colombeau and non-Colombeau regimes

Abstract

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.
Paper Structure (6 sections, 7 theorems, 112 equations)

This paper contains 6 sections, 7 theorems, 112 equations.

Table of Contents

  1. Introduction and background
  2. Set-up and main results
  3. Resolvent convergence
  4. Proof of Theorems \ref{['thm:main1']} and \ref{['thm:Colombeau-case']}
  5. The Colombeau algebra and the notions of generalised solutions and compatibility for Schrödinger equations.
  6. The operator $\Delta_\alpha$

Key Result

Theorem 2.1

When and all three equations eq:LS-NLS-Delta-eq:LS-NLS-eps are globally well-posed in $L^2(\mathbb{R}^3)$. In particular, for given initial datum in $L^2(\mathbb{R}^3)$ at $t=0$ they all admit a unique strong $L^2$-solution preserving the $L^2$-norm at all times.

Theorems & Definitions (19)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 9 more