On second order linear differential equations with inverse square singularities

We study the differential equation y″ + (a/x 2 + q(x))y = Ey where a, E ∈ ℂ and where q is a complex-valued function which is locally integrable on ℝ - {0} and analytic in a neighborhood of x = 0 (or nearly so). In particular, we are interested in the behavior of solutions of initial value problems as the initial point varies and in the asymptotic behavior of solutions as the parameter E tends to infinity.