On the constructive solution of an inverse Sturm-Liouville problem

TL;DR

This paper establishes necessary and sufficient conditions for sequences to be the spectrum and norming constants of a Sturm-Liouville boundary value problem with a real integrable potential.

Contribution

It provides a constructive solution to the inverse Sturm-Liouville problem, characterizing spectral data for a specific boundary value problem.

Findings

Conditions for spectral sequences are explicitly characterized.

The results apply to potentials in L^1 space.

A constructive method for the inverse problem is developed.

Abstract

The necessary and sufficient conditions are found for the two sequences $\left\{\mu_n \right\}_{n=0}^{\infty}$ and $\left\{a_n \right\}_{n=0}^{\infty}$ to be the spectrum and the norming constants respectively, for a boundary value problem $-y'' + q \left( x \right) y = \mu y,$ $y \left(0\right)=0,$ $y\left( \pi \right)\cos \beta + y'\left( \pi \right)\sin \beta = 0,$ $\beta \in \left( 0, \pi \right),$ with $q \in L_{\mathbb{R}}^1 \left[0, \pi \right].$