Stability of the differential equation

Let $q^ { {\rm *} \, }$ denote the Hermitian conjugate of q. For equation (80) to be stable the energy $q^ { {\rm *} \, } q$ must be either constant or decaying during depth extrapolation.

$$\begin{eqnarray} {d\ \over dz }\ ( q^ { {\rm *} \, } q) \ \ \ &\le&\ \ \ 0 \nonu... ...q^ { {\rm *} \, } q_z \ +\ q_z^ { {\rm *} \, } q\ \ \ &\le&\ \ \ 0\end{eqnarray}$$ (82)

Substituting equation (80) into equation (82) gives

$$\begin{eqnarray} q^{\rm *}\,R\,q\ +\ q^{\rm *} R^{\rm *} \, q\ \ \ &\ge&\ \ \ 0 ... ... q^ { {\rm *} \, } ( R\ +\ R^ { {\rm *} \, } ) q\ \ \ &\ge&\ \ \ 0\end{eqnarray}$$ (83)

Equation (83) shows that $R\ +\ R^ { {\rm *} \, }$must be positive semidefinite for the differential equation to be stable.