Quantum LC circuits with charge discreteness: Normal and anomalous spectrum

A solvable quantum LC circuit with charge discreteness is studied. Two discrete spectral branches arc obtained: i) the normal branch corresponding to a charged capacitor with integer effective charge k = q_en (q _e elementary charge, n integer) and ii) the anomalous branch where the energy is related to non-integer effective charge k = q_e(n - x) in the capacitor. For usual mesoscopies data like quantum point contact we found x ∼ 0.7.