Counterexamples to L^p boundedness of wave operators for classical and higher order Schrödinger operators

We consider the higher order Schrödinger operator H=(−Δ)^m+V(x) in n dimensions with real-valued potential V when n>4m−1, m∈N. We show that for any [Formula presented] and [Formula presented], there exists a real-valued, compactly supported potential V∈C^α(Rⁿ) for which the wave operators W^± are not bounded on L^p(Rⁿ). As a consequence of our analysis we show that the wave operators for the usual second order Schrödinger operator −Δ+V are unbounded on L^p(Rⁿ) for n>3 and [Formula presented] for insufficiently differentiable potentials V, and show a failure of L^p′→L^p dispersive estimates that may be of independent interest.