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Dispersive estimates for the three-dimensional Schrödinge equation with rough potentials
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 128, Number 3, June 2006
- pp. 731-750
- 10.1353/ajm.2006.0025
- Article
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The three-dimensional Schrödinger propogator eitH, H = -Δ + V, is a bounded map from L1 to L∞ with norm controlled by |t|-3/2 provided the potential satisfies two conditions: An integrability condition limiting the singularities and decay of V, and a zero-energy spectral condition on H. This is shown by expressing the spectral measure of H in terms of its resolvents and proving a family of Lp mapping estimates for the resolvents. Previous results in this direction had required V to satisfy explicit pointwise bounds.