Norm conservation for the scalar parabolic wave equation (${\partial }_x\int |U{|}^{2}dS=0$)

For $U_x=iL(x)U$ with a self-adjoint transverse differential operator $L(x)$, differentiating the squared L2 norm gives $i⟨U,LU⟩-i⟨LU,U⟩=0$. 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.