CONSTRUCTION OF EIGENFUNCTION EXPANSIONS BY THE PERTURBATION METHOD AND ITS APPLICATION TO N-DIMENSIONAL SCHRODINGER OPERATIONS.

An operator-theoretical approach to the construction of eigenfunction expansions by the perturbation method is discussed. Eigenfunction expansions in this paper mean those associated with continuous spectra. The method is to combine a stationary approach to the scattering theory with the now well developed idea of formulating generalized eigenfunction expansions in terms of a triplet of spaces. Given an expansion associated with the unperturbed operator, two complete sets of eigenfunctions of the perturbed operator are constructed and shown to be the unique solution of the Lippmann-Schwinger equation in a modified form. As an application, spectral properties of the n-dimensional Schrodinger operator -delta fx are investigated in detail.