KdV hierarchy via Abelian coverings and operator identities

Abstract

We establish precise spectral criteria for potential functions $V$ of reflectionless Schrödinger operators $L_V=-{\partial }_x^2+V$ to admit solutions to the Korteweg de-Vries (KdV) hierarchy with $V$ as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Riemann surfaces by defining generalized Abelian integrals and analogues of the Baker-Akhiezer function on infinitely connected domains with a uniformly thick boundary satisfying a fractional moment condition.

52 pages, 1 figure