  Recent novel mesoscopic two-arm experiments involving quantum dots, electron
interferometry and Aharononov-Bohm effects have enabled measuring the electron
transmission probabilities and the phases. Unexpected features in the phases as
function of the gap voltage U have simulated intensive theoretical works. It is
shown in this paper that the phases (f) and conductances (|C|), appearing in
both the experimental and the theoretical works, are interrelated through
integral expressions, causing f and log(|C|) to be Hilbert transforms. The
empirically found interrelations imply remarkable analytical properties of the
U-dependence of wave-functions in mesoscopic systems.
