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The Lamé wave equation. (English) Zbl 0715.34008

Separation of the three-dimensional Helmholtz equation $\left({\Delta }+{k}^{2}\right)u=0$ in elliptic coordinates leads to the Lamé wave equation

$\left(1\right)\phantom{\rule{1.em}{0ex}}\left[{f}^{1/2}\left(z\right)\frac{d}{dz}{f}^{1/2}\left(z\right)\frac{d}{dz}+\frac{1}{4}q\left(z\right)\right]w=0$

where $f\left(z\right)=\left(z-{a}_{1}\right)\left(z-{a}_{2}\right)\left(z-{a}_{3}\right),\phantom{\rule{1.em}{0ex}}q\left(z\right)=h-\ell z+{k}^{2}{z}^{2},$ h,$\ell$ separation constants. In § 2, the spectrum of equation (1) and the Lamé wave functions are defined and a survey on known results is given. The Klein-Bocher classification of linear second order equations with rational coefficients is presented and the position of equation (1) is indicated. In § 3, the asymptotics of the solutions in the complex z-plane are studied for arbitrary complex parameter values k,h,$\ell$. In § 4, the asymptotics of the spectrum and of the angle wave functions are studied by passing to the complex domain. For the spectral parameters h,$\ell$ a system of equations ${F}_{j}\left(h,\ell \right)=\pi {h}_{j}+{b}_{j}+o\left(1\right)$, $j=1,2$, is obtained where the ${h}_{j}$ are integers which are generalizations of the classical Bohr-Sommerfeld quantization law. The functions ${F}_{j}$ are periods of the hyperelliptic integral $S\left(z\right)={\int }_{{z}_{0}}^{z}{\left[q\left(t\right)/f\left(t\right)\right]}^{1/2}dt$.

##### MSC:
 34M99 Differential equations in the complex domain 34B05 Linear boundary value problems for ODE 33E10 Lamé, Mathieu, and spheroidal wave functions