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 supλ→∞ ∥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 αℛd,δ(λ2) f∥B* ≤ Cnλ -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 Cn 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.
|Original language||English (US)|
|Number of pages||36|
|State||Published - Jul 2009|
ASJC Scopus subject areas
- Applied Mathematics