Michael Goldberg

Dispersive Estimates for Schrödinger Operators in Dimensions One and Three

We consider L1→L∞ estimates for the time evolution of Hamiltonians H=−Δ+V in dimensions d=1 and d=3 with bound We require decay of the potentials but no regularity. In d=1 the decay assumption is ∫(1+|x|)|V(x)|dx<∞, whereas in d=3 it is |V(x)|≤C(1+|x|)−3−.

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 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 + i0))–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) ∥Dα ℛ d,δ (λ 2)f∥ B* ≤ Cnλ –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 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 Hörmanders variable coefficient Plancherel theorem for oscillatory integrals.

Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials

MICHAEL GOLDBERG, LUIS VEGA, AND NICOLA VISCIGLIAAbstract. In each dimension n ≥ 2, we construct a class of nonnegativepotentials that are homogeneous of order −σ, chosen from the range 0 ≤ σ < 2,and for which the perturbed Schro¨dinger equation does not satisfy global intime Strichartz estimates.

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in R3

We present a novel approach for bounding the resolvent of a Schroedinger operator with a large first-order perturbation. It is shown here that despite the size of the perturbation, its associated Born series is absolutely convergent at sufficiently high energies. This requires suitable smoothness and polynomial decay of the scalar and vector (magnetic) potentials. Such a result is well known in the scalar case, but is more difficult for vector potentials because the gradient term exactly cancels the natural decay of the free resolvent. To control the individual terms of the Born series, we introduce a conical decomposition of the resolvent and then sum over all possible combinations of cones. Chains of cones that all point in the same direction lead to a factorial gain typical of iterated Volterra-type operators. On the other hand, cones that are not aligned contribute little due to the decay of Fourier transforms. We make no use of micro-local analysis, but instead rely only on classical phase-space techniques. As a corollary, we show that the time evolution of the Schroedinger operator in

A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions

R^3

Schrödinger Dispersive Estimates for a Scaling-Critical Class of Potentials

satisfies global Strichartz and smoothing estimates without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance for this application.

Strichartz estimates and maximal operators for the wave equation in R3

AbstractIn dimension n > 3 we show the existence of a compactly supported potential in the differentiability class

A Dispersive Bound for Three-Dimensional Schrödinger Operators with Zero Energy Eigenvalues

C^\alpha, \alpha < \frac{n-3} 2