of $x$. In this paper, we restrict to the case where they are constants independent of $x$. In that case, the left-hand side of Eq. (\[eq:iie1\]) approaches asymptotically to a constant, $1/(k\xi)$, in the $l\rightarrow \infty$ limit. Therefore we need to calculate $Z_1$ ($=\langle r\rangle$) and $Z_2$ ($=\langle r^2\rangle$) in the $l\rightarrow \infty$ limit to obtain the localization length.

To calculate $Z_1$ and $Z_2$ for use in Eq. (\[eq:iie1\]), we derive an infinite number of coupled nonrandom differential equations satisfied by the moments $Z_{n}$, where $n$ is an arbitrary nonnegative integer, using the first of Eq. (\[eq:rt\]) and Novikov’s formula. These equations turn out to take the form $$\begin{aligned}
&&\frac{1}{k}\frac{ dZ_n}{dl}=\big[2in\left(\cos\theta-u_0\sec\theta-a_0\tan\theta\right)\nonumber\\
&&~~ +g_u n^2\left(1-3\sec^2{\theta}\right)-g_a n^2\left(1+3\tan^2{\theta}\right)\big]Z_n\nonumber\\
&&~~ -ne^{i\theta}\left[C_0-i(2n+1)C_2\right]Z_{n+1}\nonumber\\
&&~~ +ne^{-i\theta}\left[C_0-i(2n-1)C_2\right]Z_{n-1}\nonumber\\
&&~~ +\frac{1}{2}n(n+1)e^{2i\theta}C_1Z_{n+2}\nonumber\\
&&~~ +\frac{1}{2}n(n-1)e^{-2i\theta}C_1Z_{n-2}.
\label{eq:iie2}\end{aligned}$$ If the potentials are zero in $z>L$ and $z<0$, the initial conditions for $Z_{n}$’s are $Z_{0}=1$ and $Z_{n}(l=0)=0$ for $n>0$. In the $l\rightarrow \infty$ limit, all $Z_{n}$’s become independent of $l$ and the left-hand sides of these equations vanish. Then we obtain an infinite number of coupled $\it algebraic$ equations. The moments $Z_{n}$ with $n > 0$ are coupled to one another and their magnitudes decrease rapidly as $n$ increases. Based on this observation, we solve 