# §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 $\Real$; otherwise the equation is called unstable. If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\leq\mathop{a^{0}_{\nu}\/}\nolimits\!\left(k^{2}\right)$ or $h$ lies in one of the closed intervals with endpoints $\mathop{a^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$ and $\mathop{b^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, $m=1,2,\dots$. If $\nu$ is a nonnegative integer, then (29.2.1) is unstable iff $h\leq\mathop{a^{0}_{\nu}\/}\nolimits\!\left(k^{2}\right)$ or $h\in[\mathop{b^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right),\mathop{a^{m}_{\nu}\/% }\nolimits\!\left(k^{2}\right)]$ for some $m=1,2,\dots,\nu$.