KDV HIERARCHY VIA ABELIAN COVERINGS AND OPERATOR IDENTITIES

Eichinger, B. and Vandenboom, T. and Yuditskii, P. (2019) KDV HIERARCHY VIA ABELIAN COVERINGS AND OPERATOR IDENTITIES. Transactions of the American Mathematical Society Series B, 6. pp. 1-44. ISSN 2330-0000

Abstract

We establish precise spectral criteria for potential functions V of reflectionless Schrödinger operators LV = −∂x2 + V to admit solutions to the Korteweg–de Vries (KdV) hierarchy with V as an initial value. More gener-ally, 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.