  Let H be a Schrodinger operator with barrier potential on the real line. We
define the Besov spaces for H by developing the associated Littlewood-Paley
theory. This theory depends on the decay estimates of the spectral operator in
the high and low energies. We also prove a Mikhlin-Hormander type multiplier
theorem on these spaces, including the Lp boundedness result. Our approach has
potential applications to other Schrodinger operators with short-range
potentials, as well as in higher dimensions.
