On the L-P boundedness of wave operators for two-dimensional Schrodinger operators with threshold obstructions

摘要

Let H = -Delta + V be a Schrodinger operator on L-2(R-2) with real-valued potential V, and let H-0 = -Delta. If V has sufficient pointwise decay, the wave operators W-+/- = s - lim(t -+/-infinity)e(itH) e(-itH0) are known to be bounded on L-P(R-2) for all 1 p infinity 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-P(R-2) for 1 p infinity. 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. (C) 2017 Elsevier Inc. All rights reserved.