DUAL EQUATION AND INVERSE PROBLEM FOR AN INDEFINITE STURM–LIOUVILLE PROBLEM WITH M TURNING POINTS OF EVEN ORDER

Hamidreza Marasi,Aliasghar Jodayree Akbarfam

doi:10.3846/13926292.2012.732972

Hamidreza Marasi, Aliasghar Jodayree Akbarfam

Open Access

https://doi.org/10.3846/13926292.2012.732972

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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 &lt; x 1 &lt; x 2 &lt; … &lt; x m &lt; 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.

Highlights

In most differential equations with variable coefficients it is impossible to obtain an exact solution, so one must resort to various approximation methods of solution
In most papers the inverse problem was studied in the case that the weight function φ2(x) does not change sign
Many further applications connect with differential equation of the form (2.1) with turning points

Summary

Introduction

In most differential equations with variable coefficients it is impossible to obtain an exact solution, so one must resort to various approximation methods of solution. In physics, engineering and quantum mechanics it is necessary to build a mathematical model to represent certain problems, e.g. the determining of the inter-atomic forces for given energy-levels. Given the spectrum of a second-order differential operator, to determine this operator This is known as the inverse problem. Marchenko in [17] has shown that the problem can only have a solution if the spectral distribution function σ(λ) of the operator is given. Levitan and Gasymov in [13] gave a new account of the solution of the inverse problem in terms of the spectral function in the case a classical Sturm–Liouville operator. In this paper we proceed to solve such inverse problem in a case where the weight function (the coefficient of λ in (1.3)) has m turning points that one of odd order and others are of even order

Notations and Preliminary Results

Dual Equation

Inverse Problem

Conclusions