A Class of Sum Rules with Application to Nondegenerate Perturbation Theory
PETER T. LANDSBERG, David John Morgan · Journal of Mathematical Physics · 1966
The main results of this paper are: (1) The derivation of a class of sum rules. These rules are then applied to operators H(α), which depend on a parameter α and have eigenvalues En(α). Writing An(α) ≡ H(α) − En(α), these rules give the matrix elements of ∂NAn(α)/∂αN with respect to the eigenfunctions of H(α) in terms of matrix elements of lower derivatives and the eigenvalues En(α). (2) An explicit series expression for the Nth-order eigenvalue correction, due to a perturbation, directly in terms of the unperturbed eigenfunctions and eigenvalues. This expression seems a convenient one to use, since it does not involve eigenfunction corrections. An expression is also obtained for the eigenfunction corrections, but this is in less convenient form.