Thoughts about potentials with finite-band spectrum and finite-dimensional reductions of integrable systems

Abstract

We repeat, using methods developed for BKM systems, the famous results of S. Novikov (1974), J. Moser (1981, 1982) , and A. Veselov (1980) that relate Schrödinger-Hill operators with finite-band spectra, solutions of the Neumann system, and certain solutions of the KdV equations. Our general motivation is to determine whether it is possible to apply inverse scattering methods to BKM systems, and in the conclusion, we indicate initial observations in this direction.

19 pages; the paper does not contain new results but aims on understanding results of the greats using different approach, which can possibly be applied for a much wider class of integrable PDE systems. We changed the title: the title of the v1-version was "Potentials with finite-band spectrum and finite-dimensional reductions of BKM systems''