On Hill's Equation with a Singular Complex-Valued Potential

In this paper Hill's equation y' + qy = Ey, where q is a complex-valuedfunction with inverse square singularities, is studied. Resultson the dependence of solutions to initial value problems onthe parameter E and the initial point x₀, on the structure ofthe conditional stability set, and on the asymptotic distributionof (semi-)periodic and Sturm-Liouville eigenvalues are obtained.It is proved that a certain subset of the set of Floquet solutionsis a line bundle on a certain analytic curve in C². We establishnecessary and sufficient conditions for q to be algebro-geometric,that is, to be a stationary solution of some equation in theKorteweg-de Vries (KdV) hierarchy. To do this a distinctionbetween movable and immovable Dirichlet eigenvalues is employed.Finally, an example showing that the finite-band property doesnot imply that q is algebro-geometric is given. This is in contrastto the case where q is real and non-singular. 1991 MathematicsSubject Classification: 34L40, 14H60.