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

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.