  Understanding the charging of exceptionally narrow levels in quantum dots in
the presence of interactions remains a challenge within mesoscopic physics. We
address this fundamental question in the generic model of a narrow level
capacitively coupled to a broad one. Using bosonization we show that for
arbitrary capacitive coupling charging can be described by an analogy to the
magnetization in the anisotropic Kondo model, featuring a low-energy crossover
scale that depends in a power-law fashion on the tunneling amplitude to the
level. Explicit analytical expressions for the exponent are derived and
confirmed by detailed numerical and functional renormalization-group
calculations.
