§29.9 Stability

The Lamé equation (29.2.1) with specified values of $k,h,\nu$ is called stable if all of its solutions are bounded on $ℝ$; otherwise the equation is called unstable. If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h$ lies in one of the closed intervals with endpoints $a_{\nu }^m\left(k^2\right)$ and $b_{\nu }^m\left(k^2\right)$, $m=1,2,\mathrm{\dots }$. If $\nu$ is a nonnegative integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h\in [b_{\nu }^m\left(k^2\right),a_{\nu }^m\left(k^2\right)]$ for some $m=1,2,\mathrm{\dots },\nu$.