On Hill's equation with a singular complex-valued potential

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 $x_0$, 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 ${\Bbb C}^2$. 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.