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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 (*)[d 2 /dz 2 -n(n+1)k 2 sn 2 z+h]f=0, (n>0, integer), by a factorization of the operator. Defining Δ,Δ ˜=d/dz±T(z) as matrix operators, the product Δ ˜Δ 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 ' =iϵ jk T j T 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 k0, (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
MSC:
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