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In this paper we consider Schrödinger operators H = –Δ + i ( A · ∇ + ∇ · A ) + V = –Δ + L in ℝ n , 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 – ( λ 2 + i 0)) –1 is not small in operator norm on weighted L 2 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 α ℛ d,δ ( λ 2 ) f ∥ B* ≤ C n λ –1+| α | ∥ f ∥ B where 0 ≤ | α | ≤ 2, B is the Agmon-Hörmander space, and ℛ d,δ ( λ 2 ) is the free resolvent operator at energy λ 2 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.
Forum Mathematicum – de Gruyter
Published: Jul 1, 2009
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