Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions
-
M. Burak Erdoğan
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 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) ∥Dαℛd,δ(λ2)f∥B* ≤ Cnλ–1+|α|∥f∥B
where 0 ≤ |α| ≤ 2, B is the Agmon-Hörmander space, and ℛd,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.
© de Gruyter 2009
Articles in the same Issue
- The Noether Map I
- A general notion of algebraic entropy and the rank-entropy
- Computing the maximal algebra of quotients of a Lie algebra
- Mahler measure under variations of the base group
- Extremal α-pseudocompact abelian groups
- Group algebras whose symmetric and skew elements are Lie solvable
- Walks on graphs and lattices – effective bounds and applications
- Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions
- On the homotopy type of the non-completed classifying space of a p-local finite group
Articles in the same Issue
- The Noether Map I
- A general notion of algebraic entropy and the rank-entropy
- Computing the maximal algebra of quotients of a Lie algebra
- Mahler measure under variations of the base group
- Extremal α-pseudocompact abelian groups
- Group algebras whose symmetric and skew elements are Lie solvable
- Walks on graphs and lattices – effective bounds and applications
- Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions
- On the homotopy type of the non-completed classifying space of a p-local finite group
