Friedel phase discontinuity and bound states in the continuum in quantum dot systems

n this article we study the Friedel phase of the electron transport in two different systems of quantum dots which exhibit bound states in the continuum (BIC). The Friedel phase jumps abruptly in the energies of the BICs, which is associated to the vanishing width of these states, as shown by Friedrich and Wintgen in Phys. Rev. A \textbf{31}, 3964 (1985). This odd behavior of the Friedel phase has consequences in the charge through the Friedel sum rule. Namely, if the energy of the BIC drops under the Fermi energy the charge changes abruptly in a unity. We show that this behavior closely relates with discontinuities in the conductance predicted for interacting quantum dot systems.