Abstract
In this paper Hill's equation y″ + qy = Ey, where q is a complex-valued function with inverse square singularities, is studied. Results on the dependence of solutions to initial value problems on the parameter E and the initial point x0, on the structure of the conditional stability set, and on the asymptotic distribution of (semi-)periodic and Sturm-Liouville eigenvalues are obtained. It is proved that a certain subset of the set of Floquet solutions is a line bundle on a certain analytic curve in C2. We establish necessary and sufficient conditions for q to be algebro-geometric, that is, to be a stationary solution of some equation in the Korteweg-de Vries (KdV) hierarchy. To do this a distinction between movable and immovable Dirichlet eigenvalues is employed. Finally, an example showing that the finite-band property does not imply that q is algebro-geometric is given. This is in contrast to the case where q is real and non-singular. 1991 Mathematics Subject Classification: 34L40, 14H60.
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