Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in ℝ³

We present a novel approach for bounding the resolvent of H = -Δ + i (A · ∇ + ∇ · A) + V =: -Δ + L (1) for large energies. It is shown here that there exist a large integer m and a large number λ₀ so that relative to the usual weighted L²-norm, ||(L(-Δ + (λ + i0))^-1)^m|| <1/2 (2) for all λ > λ₀. This requires suitable decay and smoothness conditions on A, V. The estimate (2) is trivial when A = 0, but difficult for large A since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), 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 Volterra-type gain of the form (m!)^-ε with ε > 0 fixed. On the other hand, cones that are not aligned contribute little due to the assumed decay of Â. We make no use of micro-local analysis, but instead rely on classical phase space techniques. As a corollary of (2), we show that the time evolution of the operator in ℝ³ satisfies global Strichartz and smoothing estimates without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance.