The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

Abstract

We consider a scaling limit of a nonlinear Schrödinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equation \begin{equation*} i\frac{\partial }{\partial t} ψ^\varepsilon(t)= -Δψ^\varepsilon(t) + g(\varepsilon,μ,|(ρ^\varepsilon,ψ^\varepsilon(t))|^{2μ}) (ρ^\varepsilon,ψ^\varepsilon(t)) ρ^\varepsilon \end{equation*} where ${\rho }^{\epsilon }\to {\delta }_0$ weakly and the function $g$ embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics. The limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well-posedness, blow-up and asymptotic properties of solutions. Our result is the first justification of the model as the point limit of a regularized dynamics.

34 pages. Major changes in the introduction, updated references, corrected several minor misprints

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• Blow-up for the pointwise NLS in dimension two: absence of critical power
• The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions
• On Global attraction to solitary waves for Klein-Gordon equation with concentrated nonlinearity
• Well-posedness of the Two-dimensional Nonlinear Schrödinger Equation with Concentrated Nonlinearity
• Two-dimensional Time-dependent Point Interactions
• Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one
• Time dependent delta-prime interactions in dimension one
• A general review on the NLS equation with point-concentrated nonlinearity
• On global well-posedness for Klein-Gordon equation with concentrated nonlinearity
• The 1-D Dirac equation with concentrated nonlinearity
• Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one
• The Harmonic Oscillator Potential Perturbed by a Combination of Linear and Non-linear Dirac Delta Interactions with Application to Bose-Einstein Condensation
• Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction
• Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
• An introduction to the two-dimensional Schrödinger equation with nonlinear point interactions