L-p-boundedness of the wave operator for the one dimensional Schrodinger operator

摘要

Given a one dimensional perturbed Schrodinger operator H = - d(2)/dx(2) + V(x), we consider the associated wave operators W+/-, defined as the strong L-2 limits lim(s-->+/-infinity) e(is)He(-isH0). We prove that W+/- are bounded operators on L-p for all 1 < p < infinity, provided ( 1 + | x|)V-2(x) is an element of L-1, or else ( 1 + | x|) V( x) is an element of L-1 and 0 is not a resonance. For p = infinity we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.