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

Abstract

We present a novel approach for bounding the resolvent of a Schroedinger operator with a large first-order perturbation. It is shown here that despite the size of the perturbation, its associated Born series is absolutely convergent at sufficiently high energies. This requires suitable smoothness and polynomial decay of the scalar and vector (magnetic) potentials.

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.

As a corollary, we show that the time evolution of the Schroedinger operator in $R^3$ satisfies global Strichartz and smoothing estimates without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance for this application.