Communications in Mathematical Physics

, Volume 259, Issue 2, pp 475–509

Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue


DOI: 10.1007/s00220-005-1375-9

Cite this article as:
Yajima, K. Commun. Math. Phys. (2005) 259: 475. doi:10.1007/s00220-005-1375-9


Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in Lp spaces of scattering solutions eitHPcu, 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 ||eitHPcu||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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsGakushuin UniversityTokyoJapan