Eigenfunction expansions associated with the one‐dimensional Schroedinger operator

Abstract

AbstractWe consider the one‐dimensional Schroedinger operator H on L2(−∞, ∞) associated with where q(r) : R → R is locally integrable, λ ∈ R is a spectral parameter, and the differential expression is in Weyl's limit point cases at both endpoints. Starting from the well known Weyl‐Kodaira formulation of the eigenfunction expansion associated with H [1], an alternative formulation can be derived in such a way that the principal features contributing to it, namely a scalar spectral density function, the location and multiplicity of the spectrum, and the spectral types of the generalised eigenfunctions, can be explicitly exhibited in the resulting expansion. The proof is based on classical work of Kac [2] and the theory of subordinacy [3], [4]. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)