= Virial identity
For a sufficiently regular solution of the <mass-critical focusing nonlinear Schrödinger equation> with finite variance, the variance and its first <derivative> are
$$
I(t)=\int_{\mathbb R^d}|x|^2|u|^2\,dx,\qquad
I'(t)=4\operatorname{Im}\int_{\mathbb R^d}x\cdot\nabla u\,\overline u\,dx.
$$
A second differentiation and <integration by parts> give the <virial identity>
$$
I''(t)=16E(u),\qquad E(u)=\frac12\int|\nabla u|^2-\frac1{2+4/d}\int|u|^{2+4/d}.
$$
These constants use Laplacian coefficient one in the evolution equation. <Energy conservation> makes the second <derivative> constant. If the <energy> is negative, the resulting quadratic expression for $I(t)$ eventually becomes negative, contradicting its definition as a nonnegative <integral>; this proves <negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation> under the finite-variance hypotheses.
The weighted moment condition is additional to an <H1 norm> bound. A <localized virial identity> replaces the unbounded weight by a smooth cutoff when only tail control is available.
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