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 Schrödinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach to resonances (generalized eigenvalues) for both classes of potentials without using any spectral deformation techniques. Generalized eigenvalues for one-dimensional Schrödinger operators (ordinary differential operators) are investigated in detail. A certain holomorphic function D(λ) is constructed so that D(λ) = 0 if and only if λ is a generalized eigenvalue. It is proved that D(λ) is equivalent to the analytic continuation of the Evans function. In particular, a new formulation of the Evans function and its analytic continuation is given.