Spectral multipliers for Schrödinger operators

Abstract

We prove a sharp Hörmander multiplier theorem for Schrödinger operators H = −∆ + V on R n . The result is obtained under certain condition on a weighted L ∞ estimate, coupled with a weighted L 2 estimate for H, which is a weaker condition than that for nonnegative operators via the heat kernel approach. Our approach is elaborated in one dimension with potential V belonging to certain critical weighted L 1 class. Namely, we assume that (1 + |x|)|V (x)|dx is finite and H has no resonance at zero. In the resonance case we assume (1 + |x| 2 )|V (x)|dx is finite.

Hardy inequality and fractional Leibnitz rule for perturbed Hamiltonians on the line

“…Proof. The results in [4], [17], [1], [3], [21] imply the existence and continuity of the wave operators in L p , 1 < p < ∞, so one can deduce Bernstein inequality…”

“…Lemma 5.3. (see [3], [21]) If ϕ ∈ C ∞ 0 (R) obeys (2.6), (2.7) and V ∈ L 1 γ (R), γ = 1 + s, s ∈ (0, 1), then for M ∈ (0, ∞) the filtered Fourier transforms…”

On homogeneous Besov spaces for 1D Hamiltonians without zero resonance

“…Some basic properties of these Besov spaces and the independence of the Besov space of the choice of the Paley-Littlewood function ϕ can be found in [13].…”