  We discuss the minimal conditions for wave function spectroscopy, in which
resonant tunneling is the measurement tool. Two systems are addressed: resonant
tunneling diodes, as a toy model, and open quantum dots. The toy model is used
to analyze the crucial tunning between the necessary resolution in
current-voltage characteristics and the breakdown of the wave functions probing
potentials into a level splitting characteristic of double quantum wells. The
present results establish a parameter region where the wavefunction
spectroscopy by resonant tunneling could be achieved. In the case of open
quantum dots, a breakdown of the mapping condition is related to a change into
a double quantum dot structure induced by the local probing potential. The
analogy between the toy model and open quantum dots show that a precise control
over shape and extention of the potential probes is irrelevant for wave
function mapping. Moreover, the present system is a realization of a tunable
Fano system in the wave function mapping regime.
