Geometric & Functional Analysis GAFA

, Volume 16, Issue 3, pp 517–536

Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials

Original Paper

DOI: 10.1007/s00039-006-0568-5

Cite this article as:
Goldberg, M. GAFA, Geom. funct. anal. (2006) 16: 517. doi:10.1007/s00039-006-0568-5


We prove a dispersive estimate for the time-independent Schrödinger operator H  =   − Δ  + V in three dimensions. The potential V(x) is assumed to lie in the intersection
$$ L^{p} ({\user2{\mathbb{R}}}^{3} ) \cap L^{q} ({\user2{\mathbb{R}}}^{3} ), $$
p < 3/2 < q, and also to satisfy a generic zero-energy spectral condition. This class, which includes potentials that have pointwise decay
$$ {\left| {V(x)} \right|} \leq C(1 + {\left| x \right|})^{{ - 2 - \varepsilon }} , $$
is nearly critical with respect to the natural scaling of the Laplacian. No additional regularity, decay, or positivity of V is assumed.

Keywords and phrases.

Schrödinger equation dispersive bound a.c. spectrum resolvents limiting absorption principle stationary phase 

2000 Mathematics Subject Classification.

Primary: 35Q40 Secondary: 42A20 42B15 

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

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