KdV hierarchy via Abelian coverings and operator identities

TL;DR

This paper develops spectral criteria for reflectionless Schrödinger operators to admit solutions to the KdV hierarchy, extending classical algebro-geometric methods to noncompact Riemann surfaces with infinitely connected domains.

Contribution

It introduces generalized Abelian integrals and Baker-Akhiezer functions on complex domains, broadening the scope of KdV hierarchy solutions beyond traditional settings.

Findings

Spectral criteria for potential functions V in reflectionless Schrödinger operators.

Extension of algebro-geometric solutions to noncompact Riemann surfaces.

Framework for solutions on infinitely connected domains with boundary conditions.

Abstract

We establish precise spectral criteria for potential functions $V$ of reflectionless Schr\"odinger 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.