On the B-splines effective completeness

Effective completeness of B-splines, defined as the capability of approaching completeness without compromising the positive definite character of the corresponding superposition matrix, is investigated. A general result on the limit solution of the spectrum of B-splines superposition matrices has been obtained for a large class of knots grids. The result has been tested on finite-dimensional cases using both constant and random knots spacings (uniform distribution in [0,1]). The eigenvalue distribution for random spacings is found not to exhibit any large deviation from that for constant spacings. As an example of system which takes huge advantage of a non-uniform grid of knots, we have computed few hundreds of hydrogen Rydberg states obtaining accuracy comparable to the machine accuracy. The obtained results give solid ground to the recognized efficiency and accuracy of the B-spline sets when used in atomic physics calculations.