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.

Keywords:

• turning point
• Sturm–Liouville problem
• indefinite problem
• dual equation
• inverse problem

AMS Subject Classification:

• 34E55
• 34B05
• 34E20
• 34L05
• 47E05