  For a subclass of polar-cap models based on electromagnetic cascades induced
by curvature radiation (CR) we calculate broad-band high-energy spectra of
pulsed emission expected for classical and millisecond pulsars. The spectra are
a combination of curvature and synchrotron components. The spectrum of
curvature component breaks at 150MeV, and neither its slope nor level below
this energy are compatible with phase-averaged spectra of pulsed X-ray emission
inferred from observations. Spectral properties in the combined energy range of
ROSAT and ASCA (0.1 - 10 keV) depend upon the location of cyclotron turnover
energy epsilon_ct=\hbar{e B \over m_e c} /sin(psi) in the synchrotron
component. Unlike in outer-gap models, the available range of pitch angles psi
is rather narrow and confined to low values. For classical pulsars, a gradual
turnover begins already at 1MeV, and the level of the synchrotron spectrum
decreases. At 10keV the curvature component eventually takes over, but with
photon index alpha = 2/3, in disagreement with observations. For millisecond
pulsars, the X-ray spectra are dominated by synchrotron component with alpha
\simeq 1.5, and a sharp turnover into alpha \simeq -1 at epsilon_ct \sim 100eV.
Relations of pulsed luminosity L_X to spin-down luminosity \edot are presented
for classical and millisecond pulsars. We conclude that spectral properties and
fluxes of pulsed non-thermal X-ray emission of some objects, like the Crab or
the millisecond pulsar B1821-24, pose a challenge to the subclass of polar-cap
models based on curvature and synchrotron radiation alone.
