  A systematic theory of the conductance measurements of non-invasive (weak
probe) scanning gate microscopy is presented that provides an interpretation of
what precisely is being measured. A scattering approach is used to derive
explicit expressions for the first and second order conductance changes due to
the perturbation by the tip potential in terms of the scattering states of the
unperturbed structure. In the case of a quantum point contact, the first order
correction dominates at the conductance steps and vanishes on the plateaus
where the second order term dominates. Both corrections are non-local for a
generic structure. Only in special cases, such as that of a centrally symmetric
quantum point contact in the conductance quantization regime, can the second
order correction be unambiguously related with the local current density. In
the case of an abrupt quantum point contact we are able to obtain analytic
expressions for the scattering eigenfunctions and thus evaluate the resulting
conductance corrections.
