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

AbstractThis 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

On a representation of vector continued fractions.

Vector Pade approximants to power series with vector coefficients may be calculated using the three-term recurrence relations of vector continued fractionsif formulated in the framework of Clifford algebras. We show that the numerator and denominator polynomials of these fractions take particularly simple forms which require just a few degrees of freedom in their representation. The new description also allows the calculation of ”hybrid” approximants

Calculations for double-well potentials of perturbed oscillator type in three-dimensional systems using the Hill-determinant approach

A determination of the eigenvalues for a three-dimensional system is made by expanding the potential functionV(x,y,z;Z2, λ,β)= ?Z2[x2+y2+z2]+λ {x4+y4+z4+2β[x2y2+x2z2+y2z2]}, around its minimum. In this paper the results of extensive numerical calculations using this expansion and the Hill-determinant approach are reported for a large class of potential functions and for various values of the perturbation parametersZ2, λ, and β. PACS No.:03.6