Series Development of tlie Solution of an Eigenvalue Problem.

Abstract

In view of the divergencies of the field theory, the convergence of series developments of bound eigenfunctions in powers of the coupling constant is studied. Born approximation represents a suitable development in cases where an unperturbated state can be defined and in which the energy shift due to the interaction is finite. Another series development, at fixed energy, can be deviced. This latter development leads necessarily to coefficients which represent non-regular functions in configuration space, nevertheless, those series converge to a regular function if the fixed energy coincides with an eigenvalue of the considered problem or, in cases where the energy of the systems is given, furnish an eigenvalue problem for the coupling constant.