Numerical computation and analysis of the Titchmarsh–Weyl mα(λ) function for some simple potentials

This article is concerned with the Titchmarsh–Weyl mα(λ) function for the differential equation d2y/dx2+[λ−q(x)]y=0. The test potential q(x)=x2, for which the relevant mα(λ) functions are meromorphic, having simple poles at the points λ=4k+1 and λ=4k+3, is studied in detail. We are able to calculate the mα(λ) function both far from and near to these poles. The calculation is then extended to several other potentials, some of which do not have analytical solutions. Numerical data are given for the Titchmarsh–Weyl mα(λ) function for these potentials to illustrate the computational effectiveness of the method used.