  We present a semi-classical method for determining the effective low-energy
quantum Hamiltonian of weakly anharmonic superconducting circuits containing
mesoscopic Josephson junctions coupled to electromagnetic environments made of
an arbitrary combination of distributed and lumped elements. A convenient
basis, capturing the multi-mode physics, is given by the quantized eigenmodes
of the linearized circuit and is fully determined by a classical linear
response function. The method is used to calculate numerically the low-energy
spectrum of a 3D-transmon system, and quantitative agreement with measurements
is found.
