Dual equation and inverse problem for an indefinite Sturm–Liouville problem with m turning points of even order

Abstract

In this paper the differential equation y″ + (ρ ² φ 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 ρ ² is a real parameter. Also, [0, 1] contains a finite number of zeros of φ 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.