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The Nahm equations, finite-gap potentials and Lamé functions. (English) Zbl 0639.34042
The object of this paper is to find the eigenvalues h giving rise to doubly-periodic solutions of the Lamé equation $(*)\quad [d\sp 2/dz\sp 2-n(n+1)k\sp 2sn\sp 2z+h]f=0,$ $(n>0$, integer), by a factorization of the operator. Defining $\Delta$,${\tilde \Delta}=d/dz\pm T(z)$ as matrix operators, the product ${\tilde \Delta}\Delta$ is made to be equivalent to $2n+1$ copies of (*), each with a different eigenvalue h, thus producing the $2n+1$ Lamé polynomials of degree n. The matrix T(z) has to satisfy the Nahm equations $T'\sb{\ell}=i\epsilon\sb{jk\ell}T\sb jT\sb k,$ and is found by use of representations of so(3). The case $n=1$ is given in detail. Brief consideration is given to (i) the case when n is half an odd integer, (ii) the limit as $k\to 0$, (iii) relevance to reflectionless potential in the Schrödinger equation (iv) the possibility of using the same process to obtain more general finite-gap and reflectionless potentials.
Reviewer: F.M.Arscott

34C25Periodic solutions of ODE
34L99Ordinary differential operators
33E10Lamé, Mathieu, and spheroidal wave functions
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
47A70Eigenfunction expansions of linear operators; rigged Hilbert spaces
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