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We consider scanning gate conductance microscopy of an open quantum dot thatis connected to the conducting channel using the wave function description ofthe quantum transport and a finite difference approach. We discuss theinformation contained in conductance ($G$) maps. We demonstrate that the mapsfor a delta-like potential perturbation exactly reproduce the local density ofstates for the quantum dot that is weakly coupled to the channel, i.e. when theconnection of the channel to the dot transmits a single transport mode only. Weexplain this finding in terms of the Lippmann-Schwinger perturbation theory. Wedemonstrate that the signature of the weak coupling conditions is theconductance 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 thechannel the $G$ maps resolve the local density of states only for very specificwork points with the Fermi energy coinciding with quasi-bound energy levels.

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