Strichartz and Smoothing Estimates for Schrödinger Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions

Details Strichartz and Smoothing Estimates for Schrödinger Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions Strichartz and Smoothing Estimates for Schrödinger Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions

2007-05-03

arxiv.org/abs/0705.0546v1

Mathematics

Analysis of PDEs

36 pages

Scientific paper

In this paper we consider magnetic Schr\"odinger operators in R^n, n \ge 3. Under almost optimal conditions on the potentials in terms of decay and regularity we prove smoothing and Strichartz estimates, as well as a limiting absorption principle. For large gradient perturbations the latter is not a corollary of the free case as the differentiated free resolvent does not have small operator norm on any weighted L^2 spaces. We instead show that the spectral radius of such operators decreases to zero, hence their perturbation of the identity is still invertible. The key estimates are based on an angular decomposition of the free resolvent, or rather a bound that holds uniformly for all possible angular decompositions. The proof avoids the Fourier transform and instead uses H\"ormander's variable coefficient Plancherel theorem for oscillatory integrals.

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