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Aims. We analyze whether the effects of secular perturbations, originatingfrom a substellar companion, on the dust dynamics in a debris disk can beinvestigated with spatially resolved observations.  Methods. We numerically simulated the collisional evolution of narrow andeccentric cold planetesimal belts around a star of spectral type A3V that aresecularly perturbed by a companion that orbits either closer to or farther fromthe star than the belt. Based on the resulting spatial dust distributions, wesimulated 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 thebrightness distribution varies significantly with observing wavelength, forexample between the $N$ and $Q$ band. This can be explained by the varyingrelative contribution of the emission of the smallest grains near the blowoutlimit. The orbits of both the small grains that form the halo and the largegrains close to the parent belt precess due to the secular perturbationsinduced by a companion orbiting inward of the belt. The halo, being composed ofolder grains, trails the belt. The magnitude of the trailing decreases withincreasing perturber mass and hence with increasing strength of theperturbation, a trend we recovered in synthetic maps of surface brightness byfitting ellipses to lines of constant brightness. Systems with an outerperturber do not show a uniform halo precession since the orbits of smallgrains are strongly altered. We identified features of the brightnessdistributions suitable for distinguishing between systems with a potentiallydetectable inner or outer perturber, especially with a combined observationwith JWST/MIRI in the $Q$ band tracing small grain emission and with ALMA at mmwavelengths tracing the position of the parent planetesimal belt.

Using debris disk observations to infer substellar companions orbiting within or outside a parent planetesimal belt