Local density of states in mesoscopic samples from scanning gate microscopy

We study the relationship between the local density of states (LDOS) and the conductance variation ΔG in scanning-gate-microscopy experiments on mesoscopic structures as a charged tip scans above the sample surface. We present an analytical model showing that in the linear-response regime the conductance shift ΔG is proportional to the Hilbert transform of the LDOS and hence a generalized Kramers--Kronig relation holds between LDOS and ΔG. We analyze the physical conditions for the validity of this relationship both for one-dimensional and two-dimensional systems when several channels contribute to the transport. We focus on realistic Aharonov--Bohm rings including a random distribution of impurities and analyze the LDOS-ΔG correspondence by means of exact numerical simulations, when localized states or semiclassical orbits characterize the wave function of the system.