  We analyze the nonlinear voltage dependence of elelctrochemical capacitance
for nano-scale conductors. This voltage dependence is due to finite density of
states of the conductors. We derive an exact expression for the electrochemical
capacitance-voltage curve for a parallel plate system. The result suggests a
quantum scanning capacitance microscopy at the nano-scale: by inverting the
capacitance-voltage expression one is able to deduce the local spectral
function of the nano-scale conductor.
