Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]