L^p boundedness of the wave operator for the one dimensional Schroedinger operator

TL;DR

This paper proves the boundedness of wave operators on L^p spaces for one-dimensional Schrödinger operators with certain decay conditions on the potential, extending understanding of dispersive properties in quantum mechanics.

Contribution

It establishes L^p boundedness of wave operators under specific decay conditions on the potential, including cases with and without resonances, and provides estimates at p=∞.

Findings

Wave operators are bounded on L^p for 1<p<∞ under decay conditions.

At p=∞, estimates involve the Hilbert transform.

Applications include dispersive estimates for variable coefficient equations.

Abstract

Given a one dimensional perturbed Schroedinger operator H=-(d/dx)^2+V(x) we consider the associated wave operators W_+, W_- defined as the strong L^2 limits as s-> \pm\infty of the operators e^{isH} e^{-isH_0} We prove that the wave operators are bounded operators on L^p for all 1<p<\infty, provided (1+|x|)^2 V(x) is integrable, or else (1+|x|)V(x) is integrable and 0 is not a resonance. For p=\infty we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.