On the inverse eigenvalue problem for matrices (atomic corrections)

Abstract

The inverse eigenvalue problem for matrices is studied with the objective of obtaining an efficient method for correcting energy levels in atomic systems, though the results are applicable to any eigenvalue problem. The approach is a development of earlier work by S. Friedland (1977). The diagonal elements of a real symmetric matrix with given off-diagonal elements are adjusted to yield a given spectrum. The authors discuss cases where there are real solutions and no real solutions, with particular emphasis on the latter. Problems of slow convergence arise. They demonstrate the cause of this slow convergence, give a geometrical interpretation of the problem and show how it can be avoided. Also the matrices encountered arise from complex calculations and are subject to error. They develop an error analysis that permits us, among other things, to judge whether corrections in any particular case are justified in view of anticipated errors in the given computed off-diagonal matrix elements. Finally, the method is demonstrated in an application to certain sets of levels in 12 times ionized (neon-like) titanium.