Intermediate-energy instability for the smaller mode in the stable high-energy regime

Let $m,n$ be the mode indices, let $P$ be the parameter, and define

$$I_S=\bigcup_{k\in\mathbb N}\bigl(k(2k+1),(k+1)(2k+1)\bigr).$$

Let $\Theta_m$ and $\Theta_n$ be the corresponding modes with energy $E_{\Theta_m}$. Assume

$$0\leqslant P<m^2<n^2,\qquad \frac{n^2}{m^2}\in I_S.$$

Let $E_1$ and $E_2$ be the thresholds from the preceding stability theorem. Intermediate-energy instability conjecture. There exist $0<E_1\leqslant E_3<E_4\leqslant E_2$ such that $\Theta_m$ is linearly stable with respect to $\Theta_n$ if $0<E_{\Theta_m}<E_1$ or $E_{\Theta_m}>E_2$, and is linearly unstable with respect to $\Theta_n$ if $E_3<E_{\Theta_m}<E_4$.

The theorem establishes stability at low energies and, for ratios in $I_S$, at sufficiently high energies. The conjecture predicts an instability window at intermediate energies.