Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue

Abstract

Let H=−Δ+V(x) be a three dimensional Schrodinger operator. We study the time decay in Lp spaces of scattering solutions e−itHPcu, where Pc is the orthogonal projection onto the continuous spectral subspace of L2(R3) for H. Under suitable decay assumptions on V(x) it is shown that they satisfy the so-called Lp-Lq estimates ||e−itHPcu||p≤(4π|t|)−3(1/2−1/p)||u||q for all 1≤q≤2≤p≤∞ with 1/p+1/q=1 if H has no threshold resonance and eigenvalue; and for all 3/2<q≤2≤p<3 if otherwise.