Analytic structure of the eigenvalue problem as used in semiclassical theory of electronically inelastic collisions

The analytic properties of eigenvalues and eigenvectors of analytic symmetric matrices of arbitrary size are discussed, with special attention given to the branch point structure at a degeneracy of two eigenvalues—that is, at a complex crossing point of two potential surfaces. The discussion is relevant to semiclassical theory of electronically inelastic collisions.