Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\mathbb{R}^3$

Abstract

Such a result is well known in the scalar case, but is more difficult for vector potentials because the gradient term exactly cancels the natural decay of the free resolvent. To control the individual terms of the Born series, we introduce a conical decomposition of the resolvent and then sum over all possible combinations of cones. Chains of cones that all point in the same direction lead to a factorial gain typical of iterated Volterra-type operators. On the other hand, cones that are not aligned contribute little due to the decay of Fourier transforms. We make no use of micro-local analysis, but instead rely only on classical phase-space techniques.