Spectral multipliers for Schrödinger operators

We prove a sharp Hörmander multiplier theorem for Schrödinger operators H = -Δ + V on ℝⁿ. The result is obtained under certain condition on a weighted L^∞ estimate, coupled with a weighted L² 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 L¹ 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.