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Let $\eta >0$ and let $P_{\eta }\in L^4({\mathbb{R}}^2)$ solve the trapped focusing cubic ground-state equation. In the Focusing Schrödinger lens ansatz, the initial conditions $\lambda (-1)=\sqrt{1+\eta }$ and $b(-1)=1$ giveIndeed $b=-\lambda {\lambda }_t$ and ${\lambda }^2{b}_t+b^2=-\eta$. The positive $\eta$ keeps the scale nonzero for every real time.

A change of variables $y=x/\lambda (t)$ proves the critical spacetime bound exactly:Thus the solution expands at large times and has finite $L^4$ spacetime norm, rather than merely pointwise decay. This is the norm used by scattering from a finite Strichartz norm.^[ref]

Rémi Carles studied critical nonlinear Schrödinger equations with and without a harmonic trapping potential. His work gives a lens transform relating the two evolutions in the mass-critical case. This permits information about existence and concentration in one equation to be transferred to the other, while keeping track of the transformed time interval.^[ref]

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 equationSet $\lambda (t)=\sqrt{1+t^2}$. Thensolves the Defocusing cubic nonlinear Schrödinger equation in two dimensions. Direct substitution gives the scale/chirp conditions $b=-t=-\lambda {\lambda }_t$, ${\lambda }^2{b}_t+b^2=-1$, and phase rate ${\gamma }_t=-\mu /{\lambda }^2$. The displayed stationary equation supplies exactly the remaining spatial terms.^[ref]

The harmonic-oscillator energy space assumption controls both the ordinary gradient and the quadratic-phase gradient. Moreover,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.