Explicit formula for Schrödinger wave operators on the half-line for potentials up to optimal decay

Abstract

We give an explicit formula for the wave operators for perturbations of the Dirichlet Laplacian by a potential on the half-line. The potential is assumed to decay strictly faster than the polynomial of degree minus two. The formula consists of the main term given by the scattering operator and a function of the generator of the dilation group, and a Hilbert-Schmidt remainder term. Our method is based on the elementary construction of the generalized Fourier transforms in terms of the solutions of the Volterra integral equations. The Hilbert-Schmidt property of the remainder term follows from a decay estimate for the Jost solution, which is established by performing a perturbation expansion of sufficiently high order. As a corollary, a topological interpretation of Levinson's theorem is established via an index theorem approach.