, Volume 259, Issue 2, pp 475-509
Date: 28 Jun 2005

Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue

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Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in L p spaces of scattering solutions e itH P c u, where P c is the orthogonal projection onto the continuous spectral subspace of L 2(R 3) for H. Under suitable decay assumptions on V(x) it is shown that they satisfy the so-called L p -L q estimates ||e itH P c u|| 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.

Communicated by B. Simon