A SPECTRAL THEORY OF LINEAR OPERATORS ON RIGGED HILBERT SPACES UNDER ANALYTICITY CONDITIONS II: APPLICATIONS TO SCHRÖDINGER OPERATORS

Abstract

A spectral theory of linear operators on a rigged Hilbert space is applied to Schrodinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach t o resonances (generalized eigenvalues) for both classes of potentials without using a ny spectral deformation tech- niques. Generalized eigenvalues for one dimensional Schrodinger operators (ordinary differential operators) are investigated in detail. A certain h olomorphic functionD(λ) is constructed so thatD(λ) = 0 if and only ifλ is a generalized eigenvalue. It is proved that D(λ) is equivalent to the analytic continuation of the Evans fun ction. In particular, a new formulation of the Evans function and its analytic continuation is given.