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Zbl 0715.34008
Fedoryuk, M.V.
The Lamé wave equation.
(English. Russian original)
[J] Russ. Math. Surv. 44, No.1, 153-180 (1989); translation from Usp. Mat. Nauk 44, No.1(265), 123-144 (1989). ISSN 0036-0279

Separation of the three-dimensional Helmholtz equation $(\Delta +k\sp 2)u=0$ in elliptic coordinates leads to the Lamé wave equation $$(1)\quad [f\sp{1/2}(z)\frac{d}{dz}f\sp{1/2}(z)\frac{d}{dz}+\frac{1}{4}q(z)]w=0$$ where $f(z)=(z-a\sb 1)(z-a\sb 2)(z-a\sb 3),\quad q(z)=h-\ell z+k\sp 2z\sp 2,$ h,$\ell$ separation constants. In {\S} 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 {\S} 3, the asymptotics of the solutions in the complex z-plane are studied for arbitrary complex parameter values k,h,$\ell$. In {\S} 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 $F\sb j(h,\ell)=\pi h\sb j+b\sb j+o(1)$, $j=1,2$, is obtained where the $h\sb j$ are integers which are generalizations of the classical Bohr-Sommerfeld quantization law. The functions $F\sb j$ are periods of the hyperelliptic integral $S(z)=\int\sp{z}\sb{z\sb 0}[q(t)/f(t)]\sp{1/2}dt$.
MSC 2000:
*34M99 Differential equations in the complex domain
34B05 Linear boundary value problems of ODE
33E10 Spheroidal wave functions, etc.

Keywords: Helmholtz equation; spectrum; Lamé wave functions; Klein-Bocher classification; asymptotics; angle wave functions

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