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.

26 Pages

References in corpus (2)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• L^p boundedness of the wave operator for the one dimensional Schroedinger operator

Cited by in corpus (1)

• Spectral multipliers for Schroedinger operators with Poeschl-Teller potential