Spectral Multipliers for Schrödinger Operators
Illinois Journal of Mathematics
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.
"Spectral Multipliers for Schrödinger Operators."
Illinois Journal of Mathematics, 54 (2): 621-647.