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Assume the relative permittivity is real and the field decays sufficiently at transverse infinity, or impose boundary conditions with zero transverse flux. Then $L(x)$ is self-adjoint on its transverse domain by integration by parts. The scalar parabolic wave equation givesThis is norm conservation for the scalar parabolic wave equation. It remains true when $L$ depends explicitly on $x$; its instantaneous self-adjointness suffices. Equivalently, the local balance isSince $|E|=|U|$, the invariant can also be writtenFor a unit transverse polarization in a fixed reference medium with impedance $Z_0=\sqrt{{\mu }_0/{ϵ}_0}$, the leading forward paraxial Poynting vector has $S_x\simeq |E{|}^2/(2Z_0)$. Thus the scalar model conserves its leading reference-normalized power $P_0=\mathcal{N}/(2Z_0)$. This is the intended consequence of the Hermitian-operator hint. Complex permittivity would add absorption or gain and invalidate this norm invariant.

Two qualifications are necessary for the printed physical interpretation. First, $(1,0,1)$ is neither a unit vector nor transverse to nearly axial propagation: its longitudinal electric component is as large as its transverse one. In a homogeneous source-free medium, the example $\mathbf{E}=(1,0,1)e^{ikx}$ satisfies the componentwise Helmholtz equation but violates $\nabla \cdot \mathbf{E}=0$. It therefore is not the stated near-axial electromagnetic plane wave. The same scalar calculation formally conserves $\int |\mathbf{U}{|}^2=2\mathcal{N}$ for that fixed vector, but does not make it a physical Maxwell field. A unit transverse polarization, for example $(0,0,1)$, is the consistent paraxial choice, with only small longitudinal corrections.

Second, for varying local refractive index $n(x)=\sqrt{{ϵ}_{\mathrm{rel}}(x)}$, the quoted local plane-wave impedance relation gives $Z(x)=Z_0/n(x)$ and hence a flux proportional to $n(x)|E{|}^2$, not merely $|E{|}^2$. The scalar norm proof alone cannot make this exactly constant for arbitrary $n(x)$. For example a transversely uniform envelope obeying the printed parabolic equation hasso its intensity is constant while the locally inferred plane-wave flux changes with $n$. This supplies a direct obstruction to the literal exact-flux claim within the supplied scalar assumptions.

The consistent physical statement follows from Poynting theorem: for a lossless Maxwell solution and no lateral energy leakage,For a slowly varying transverse plane wave, the corresponding forward WKB field has amplitude proportional to $n^{-1/2}$, making $n|E{|}^2$ constant. In one dimension this is also encoded in the exact conserved flux $(2{\mu }_0\omega {)}^{-1}\mathrm{Im}(E^{\ast }{E}_x)$. Thus the requested physical conservation holds for the corrected lossless Maxwell description, while the given scalar equation proves the reference-normalized paraxial invariant above. The supplied fixed longitudinal polarization and arbitrary local-impedance inference cannot be used to claim more.