KdV hierarchy via Abelian coverings and operator identities

Eichinger, B.; VandenBoom, T.; Yuditskii, P.

doi:10.1090/btran/30

KdV hierarchy via Abelian coverings and operator identities
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by B. Eichinger, T. VandenBoom and P. Yuditskii;

Trans. Amer. Math. Soc. Ser. B 6 (2019), 1-44

DOI: https://doi.org/10.1090/btran/30

Published electronically: January 2, 2019

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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.

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