ON THE Lp BOUNDEDNESS OF WAVE OPERATORS FOR TWO-DIMENSIONAL SCHRÖDINGER OPERATORS WITH THRESHOLD OBSTRUCTIONS

Let H = −∆ + V be a Schrödinger operator on L(R) with real-valued potential V , and let H0 = −∆. If V has sufficient pointwise decay, the wave operators W± = s − limt→±∞ eitHe−itH0 are known to be bounded on L(R) for all 1 < p < ∞ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on L(R) for 1 < p < ∞. This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents p.