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The Lamé wave equation. (English) Zbl 0715.34008

Separation of the three-dimensional Helmholtz equation (Δ+k 2 )u=0 in elliptic coordinates leads to the Lamé wave equation

(1)[f 1/2 (z)d dzf 1/2 (z)d dz+1 4q(z)]w=0

where f(z)=(z-a 1 )(z-a 2 )(z-a 3 ),q(z)=h-z+k 2 z 2 , h, 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,. 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, a system of equations F j (h,)=πh j +b j +o(1), 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(z)= z 0 z [q(t)/f(t)] 1/2 dt.

MSC:
34M99Differential equations in the complex domain
34B05Linear boundary value problems for ODE
33E10Lamé, Mathieu, and spheroidal wave functions