  In this article we give an overview on some recent development of
Littlewood-Paley theory for Schr\"odinger operators. We extend the
Littlewood-Paley theory for special potentials considered in the authors'
previous work. We elaborate our approach by considering potential in
$C^\infty_0$ or Schwartz class in one dimension. In particular the low energy
estimates are treated by establishing some new and refined asymptotics for the
eigenfunctions and their Fourier transforms. We give maximal function
characterization of the Besov spaces and Triebel-Lizorkin spaces associated
with $H$. Then we prove a spectral multiplier theorem on these spaces and
derive Strichartz estimates for the wave equation with a potential. We also
consider similar problem for the unbounded potentials in the Hermite and
Laguerre cases, whose potentials $V=a|x|^2+b|x|^{-2}$ are known to be critical
in the study of perturbation of nonlinear dispersive equations. This improves
upon the previous results when we apply the upper Gaussian bound for the heat
kernel and its gradient.
