Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions

Abstract

In this paper we consider Schrödinger operators

H = –Δ + i(A · ∇ + ∇ · A) + V = –Δ + L

in ℝⁿ, n ≥ 3. Under almost optimal conditions on A and V both 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 an immediate corollary of the free case as T(λ) := L(–Δ – (λ² + i0))^–1 is not small in operator norm on weighted L² spaces as λ → ∞. We instead deduce the existence of inverses (I + T(λ))^–1 by showing that the spectral radius of T (λ) decreases to zero. In particular, there is an integer m such that lim sup_λ→∞ ∥T(λ)^m∥ < . This is based on an angular decomposition of the free resolvent for which we establish the limiting absorption bound

(0.1) ∥D^αℛ_d,δ(λ²)f∥_B* ≤ C_nλ^–1+|α|∥f∥_B

where 0 ≤ |α| ≤ 2, B is the Agmon-Hörmander space, and ℛ_d,δ(λ²) is the free resolvent operator at energy λ² whose kernel is restricted in angle to a cone of size δ and by d away from the diagonal x = y. The main point is that C_n only depends on the dimension, but not on the various cut-offs. The proof of (0.1) avoids the Fourier transform and instead uses Hörmander's variable coefficient Plancherel theorem for oscillatory integrals.