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

Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

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

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References (30)

Publisher
de Gruyter
Copyright
© de Gruyter 2009
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORUM.2009.035
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Forum Mathematicumde Gruyter

Published: Jul 1, 2009

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