The analytic continuation of a group of eigenvalues

Abstract

The branch point structure of a group of eigenvalues that are functions of some parameter is used to derive a method for their analytic continuation as the solutions of an algebraic equation. The procedure is valid so long as no member of the group branches with the rest of the spectrum. The perturbation series for the total projection operator for the group is derived in a form such that small denominators due to near degeneracy are eliminated. This then allows the continuation of eigenfunctions and properties by diagonalization of the projected hamiltonian. This provides a formalism for nearly degenerate perturbation theory with clear convergence properties. Other treatments may be obtained by suitable choice of a transformation function that relates the group projector to its unperturbed value; it is shown that the function proposed by Kato has useful diabatic properties.