The potential structure formed at a sharp change in diameter of a diffusion-controlled positive column of a low-pressure and low-current dc He glow discharge has been experimentally investigated in relation to either externally or self-excited ionization waves. A strong electric field evolved on a typical distance of 1 cm in front of the constriction to provide the continuity of discharge current through it, when the constricted part of the plasma column is on the anode side. The electron energy distribution function is not in equilibrium with the field, showing the formation of a fast electron group that is quickly dumped out towards the anode because of the inelastic processes and the radial loss. Either an externally or a self-excited ionization wave disturbs the potential structure, by periodically weakening and strengthening it. As a result, large fluctuations of the local axial electric field in the structure occur and, related to them, fluctuations of the discharge voltage and current are yielded, which make the external discharge circuit a positive feedback loop for the ionization waves.