95-146 F. Gesztesy, R. Weikard

Picard Potentials and Hill's Equation on a Torus (93K, AMSLaTeX 1.1) Mar 14, 95

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Abstract. An explicit characterization of all elliptic (algebro-geometric) finite-gap solutions of the KdV hierarchy is presented. More precisely, we show that an elliptic function $q$ is an algebro-geometric finite-gap potential, i.e., a solution of some equation of the stationary KdV hierarchy, if and only if every solution of the associated differential equation $\psi''+q\psi=E\psi$ is a meromorphic function of the independent variable for every complex value of the spectral parameter $E$. Our result also provides an explicit condition for a classical theorem of Picard to hold. This theorem guarantees the existence of solutions which are elliptic of the second kind for second-order ordinary differential equations with elliptic coefficients associated with a common period lattice. The fundamental link between Picard's theorem and elliptic finite-gap solutions of completely integrable hierarchies of nonlinear evolution equations, as established in this paper, is without precedent in the literature. In addition, a detailed description of the singularity structure of the Green's function of the operator $H=d^2/dx^2+q$ in $L^2(\bbR)$ and its precise connection with the branch and singular points of the underlying hyperelliptic curve is given.

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