= Lens transform
The <lens transform> converts free <Schrödinger equation> evolution into evolution with a harmonic trapping potential by rescaling space, changing time and adding a quadratic phase. With the convention $i\partial_su=-\tfrac12\Delta u$ on $\mathbb R^d$, put
$$
v(t,x)=(\cos t)^{-d/2}e^{-i|x|^2\tan(t)/2}u\!\left(\tan t,\frac{x}{\cos t}\right),\qquad |t|<\frac\pi2.
$$
Then $i\partial_tv=-\tfrac12\Delta v+\tfrac12|x|^2v$. The prefactor preserves the <L2 space> norm under the spatial change of variables. Infinite free time $s\to\pm\infty$ is represented by finite endpoints $t\to\pm\pi/2$.<A pseudoconformal compactification of the nonlinear Schrödinger equation and applications by Terence Tao>[Section 2, definition and transformed equation]{ref}
For $i\partial_su=-\tfrac12\Delta u+\lambda|u|^{p-1}u$, the transformed nonlinear coefficient is $\lambda(\cos t)^{d(p-1)/2-2}$. It remains constant precisely at $p=1+4/d$. This calculation explains why the transform is particularly useful for the mass-critical <nonlinear Schrödinger equation>. Its time interval and normalization are essential parts of the formula.
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