Digital Library of Mathematical Functions
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29 Lamé FunctionsLamé Functions

§29.9 Stability

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Notes:
See Magnus and Winkler (1966, §2.1).
Keywords:
Lamé’s equation
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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.

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