A numerical method for the determination of the Titchmarsh-Weyl m -coefficient

Abstract

A numerical method for determining the Titchmarsh-Weyl m ( λ ) function for the singular eigenvalue equation – ( py' )' + qy = λwy on [ a ,∞), where a is finite, is presented. The algorithm, based on Weyl’s theory, utilizes a result first used by Atkinson to map a point on the real line onto the Weyl circle in the complex plane. In the limit-point case these circles ‘nest’ and tend to the limit-point m ( λ ). Using Weyl’s result for the diameter of the circles, error estimates for m ( λ ) are obtained. In 1971, W. N. Everitt obtained an extension of an integral inequality of Hardy-Littlewood, namely the help inequality. He showed that the existence of that inequality is determined by the properties of the null set of Im[ λ 2 m ( λ )]. In view of the major difficulties in analysing m ( λ ) even in the rare cases when it is given explicitly, very few examples of the help inequality are known. The computational techniques discussed in this paper have been applied to the problem of finding best constants in these inequalities.