  Aims. We analyze whether the effects of secular perturbations, originating
from a substellar companion, on the dust dynamics in a debris disk can be
investigated with spatially resolved observations.
  Methods. We numerically simulated the collisional evolution of narrow and
eccentric cold planetesimal belts around a star of spectral type A3V that are
secularly perturbed by a companion that orbits either closer to or farther from
the star than the belt. Based on the resulting spatial dust distributions, we
simulated spatially resolved maps of their surface brightness in the $K$, $N$,
and $Q$ bands and at wavelengths of 70$\mu$m and 1300$\mu$m.
  Results. Assuming a nearby debris disk seen face-on, we find that the
brightness distribution varies significantly with observing wavelength, for
example between the $N$ and $Q$ band. This can be explained by the varying
relative contribution of the emission of the smallest grains near the blowout
limit. The orbits of both the small grains that form the halo and the large
grains close to the parent belt precess due to the secular perturbations
induced by a companion orbiting inward of the belt. The halo, being composed of
older grains, trails the belt. The magnitude of the trailing decreases with
increasing perturber mass and hence with increasing strength of the
perturbation, a trend we recovered in synthetic maps of surface brightness by
fitting ellipses to lines of constant brightness. Systems with an outer
perturber do not show a uniform halo precession since the orbits of small
grains are strongly altered. We identified features of the brightness
distributions suitable for distinguishing between systems with a potentially
detectable inner or outer perturber, especially with a combined observation
with JWST/MIRI in the $Q$ band tracing small grain emission and with ALMA at mm
wavelengths tracing the position of the parent planetesimal belt.
