  Let $H$ be a Schr\"odinger operator on $\R^n$. Under a polynomial decay
condition for the kernel of its spectral operator, we show that the Besov
spaces and Triebel-Lizorkin spaces associated with $H$ are well defined. We
further give a Littlewood-Paley characterization of $L_p$ spaces as well as
Sobolev spaces in terms of dyadic functions of $H$. This generalizes and
strengthens the previous result when the heat kernel of $H$ satisfies certain
upper Gaussian bound.
