§29.9 Stability

Notes:: See Magnus and Winkler (1966, §2.1).
Keywords:: Lamé’s equation
Permalink:: http://dlmf.nist.gov/29.9

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 $m=1,2,\dots,\nu$ .