Abstract
Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in Lp spaces of scattering solutions e−itHP c u, where P c 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−itHP 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.
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Yajima, K. Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue. Commun. Math. Phys. 259, 475–509 (2005). https://doi.org/10.1007/s00220-005-1375-9
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DOI: https://doi.org/10.1007/s00220-005-1375-9