Lemma: Compact-region mode bound

For $j=\mathrm{H},\mathrm{I}$, $|x|\le X$, and $\omega >0$,

Proof. For $0\le \omega \le {\omega }_c$, use $|\sin (\omega z)/\omega |\le |z|$ in the Jost Volterra series. It gives $|f_{\pm }(x,\omega )|\le e^{m_X}$ on the compact region. Together with $|\tau |\le 4\omega$, this proves the low-frequency bound.

For $\omega \ge {\omega }_c$, the Volterra series gives

Current conservation gives $|\tau |\le 1$. This proves the remaining bound.