Dual Equation and Inverse Problem for an Indefinite Sturm-Liouville Problem with m Turning Points of Even Order

In this paper the differential equation y '' +(rho(2)phi(2)(x) - q(x))y = 0 is considered on a finite interval I, say I = [0, 1], where q is a positive sufficiently smooth function and rho(2) is a real parameter. Also, [0, 1] contains a finite number of zeros of phi(2), the so called turning points, 0 < x(1) < x(2) < ... < x(m) < 1. First we obtain the infinite product representation of the solution. The product representation, satisfies in the original equation. As a result the associated dual equation is derived and then we proceed with the solution of the inverse problem.