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

TL;DR

This paper explores the connections between finite-band spectra, integrable systems, and inverse scattering methods, revisiting classical results and proposing potential new applications for BKM systems.

Contribution

It extends classical results relating Schrödinger operators, Neumann systems, and KdV solutions to BKM systems, and discusses the possibility of applying inverse scattering techniques to these systems.

Findings

Revisited classical relations between spectral theory and integrable systems.

Indicated initial steps towards applying inverse scattering to BKM systems.

Provided insights into 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 Schr\"odinger-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.