The Lp Boundedness of Wave Operators for Schrödinger Operators with Threshold Singuralities I. The Odd Dimensional Case

Abstract. Let H = − ∆ + V (x) be an odd m-dimensional Schrödinger operator, m ≥ 3, H0 = −∆, and let W ± = lim t→± ∞ e itHe−itH0 be the wave operators for the pair (H,H0). We say H is of generic type if 0 is not an eigenvalue nor a resonance of H and of exceptional type if otherwise. We assume that V sat-isfies F(〈x〉−2σV) ∈ Lm ∗ for some σ> 1m ∗ , m ∗ = m−1m−2. We show that W ± are bounded in Lp(Rm) for all 1 ≤ p ≤ ∞ if V satisfies in addition |V (x) | ≤ C〈x〉−m−2−ε for some ε> 0 and if H is of generic type; and that W ± are bounded in Lp(Rm) for all p between mm−2 and m2 but not for p outside the closed interval [ m m−2, m 2] if V satis-fies |V (x) | ≤ C〈x〉−m−3−ε and if H is of exceptional type. This in particular implies that the continuous part of the propagator satisfies the Lp-Lq estimates ‖e−itHPc(H)u‖p ≤ C|t | 1m ( 12 − 1q)‖u‖q for the dual exponents 1p + 1 q = 1 such that 1 ≤ q ≤ 2 ≤ p ≤ ∞ if H is of generic type, and for mm−2 < q ≤ 2 ≤ p < m2, m ≥ 5, or 32 < q ≤ 2 ≤ p < 3, m = 3, if H of exceptional type. 1