Continued fraction solutions in degenerate perturbation theory

Abstract

A generalization of a previously (see ibid., vol.9, p.18-1 (1976)) derived expansion of a determinant in terms of determinants of progressively lower order is obtained and used to discuss the eigenvalue problem in the cases when two or more elements are degenerate. The development is exact for the case of a finite determinant, and represents a perturbation series for an infinite determinant. A procedure is given for writing down the expansion to any order without using detailed algebra. An application to inhomogeneous systems of linear equations is also briefly discussed. The continued fraction methods are more straightforward to apply (particularly for the higher orders) and are more rapidly convergent than standard perturbation theory.