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

Abstract

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. © European Mathematical Society 2008.