Conductance microscopy of quantum dots weakly or strongly coupled to the conducting channel

We consider scanning gate conductance microscopy of an open quantum dot that is connected to the conducting channel using the wave function description of the quantum transport and a finite difference approach. We discuss the information contained in conductance ($G$) maps. We demonstrate that the maps for a delta-like potential perturbation exactly reproduce the local density of states for the quantum dot that is weakly coupled to the channel, i.e. when the connection of the channel to the dot transmits a single transport mode only. We explain this finding in terms of the Lippmann-Schwinger perturbation theory. We demonstrate that the signature of the weak coupling conditions is the conductance which for $P$ subbands at the Fermi level varies between $P-1$ and $P$ in units of ${2e^2}/{h}$. For stronger coupling of the quantum dot to the channel the $G$ maps resolve the local density of states only for very specific work points with the Fermi energy coinciding with quasi-bound energy levels.