= Schrödinger expanding lens ansatz for the mass-critical equation
{c}
A concrete <Schrödinger expanding lens ansatz for the mass-critical equation> starts with a finite-energy solution $v\in\Sigma$ of the trapped defocusing stationary equation
$$
\left(-\Delta+\frac{|y|^2}{4}+|v|^2\right)v=\mu v,\qquad y\in\mathbb R^2.
$$
Set $\lambda(t)=\sqrt{1+t^2}$. Then
$$
u(t,x)=\lambda(t)^{-1}v\!\left(\frac{x}{\lambda(t)}\right)
\exp\!\left(\frac{it|x|^2}{4\lambda(t)^2}-i\mu\arctan t\right)
$$
solves the <Defocusing cubic nonlinear Schrödinger equation in two dimensions>. Direct substitution gives the scale/chirp conditions $b=-t=-\lambda\lambda_t$, $\lambda^2b_t+b^2=-1$, and phase rate $\gamma_t=-\mu/\lambda^2$. The displayed stationary equation supplies exactly the remaining spatial terms.<Critical nonlinear Schrödinger equations with and without harmonic potential by Rémi Carles>[Equations (1.4)–(1.5), free/trapped lens transform, rescaled to Laplacian coefficient one]{ref}
The <harmonic-oscillator energy space> assumption controls both the ordinary gradient and the quadratic-phase gradient. Moreover,
$$
\|u\|_{L^4(\mathbb R\times\mathbb R^2)}^4=\pi\|v\|_4^4,
$$
by the substitution $x=\lambda y$ and $\int_{\mathbb R}(1+t^2)^{-1}dt=\pi$. This gives an explicit free solution to which <scattering from a finite Strichartz norm> applies.
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