Spectral multipliers for Schrödinger operators

Abstract

We prove a sharp Hörmander multiplier theorem for Schrödinger operators $H=-\Delta+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ 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 $\int(1+|x|) |V(x)|\,dx$ is finite and $H$ has no resonance at zero. In the resonance case, we assume $\int(1+|x|^2) |V(x)|\, dx$ is finite.