= Norm conservation for the scalar parabolic wave equation
{title2=$\partial_x\int|U|^2\,dS=0$}
For $U_x=iL(x)U$ with a self-adjoint transverse <differential operator> $L(x)$, differentiating the squared <L2 norm> gives $i\langle U,LU\rangle-i\langle LU,U\rangle=0$. Reality of the refractive coefficient and vanishing transverse boundary flux are required. This is exact for the scalar envelope model. Its physical interpretation as leading paraxial power uses a fixed reference impedance; a varying local impedance or a longitudinal electromagnetic polarization requires a consistent Maxwell calculation and cannot be inferred from this invariant alone.
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