 95146 F. Gesztesy, R. Weikard
 Picard Potentials and Hill's Equation on a Torus
(93K, AMSLaTeX 1.1)
Mar 14, 95

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

Abstract. An explicit characterization of all elliptic (algebrogeometric)
finitegap solutions of the KdV hierarchy is presented. More precisely,
we show that an elliptic function $q$ is an algebrogeometric
finitegap 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 secondorder ordinary
differential equations with elliptic coefficients associated with a
common period lattice. The fundamental link between Picard's theorem
and elliptic finitegap 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.
 Files:
95146.tex