Low-frequency dispersive estimates for the Schrödinger group in higher dimensions

Abstract

For a large class of real-valued potentials, V(x), x∈Rⁿ, n≥4, we prove dispersive estimates for the low-frequency part of e^it(−Δ+V)P_ac, provided the zero is neither an eigenvalue nor a resonance of −Δ+V, where P_ac is the spectral projection onto the absolutely continuous spectrum of −Δ+V. This class includes potentials V∈L^∞(Rⁿ) satisfying V(x)=O(〈x〉^{−(n+2)/2−ε}), ε>0. As a consequence, we extend the results in Commun. Pure Appl. Math. 44 (1991) to a larger class of potentials.