  We analyze the spectrum of the 3-site Bose-Hubbard model with periodic
boundary conditions using a semiclassical method. The Bohr-Sommerfeld
quantization is applied to an effective classical Hamiltonian which we derive
using resonance normal form theory. The derivation takes into account the 1:1
resonance between frequencies of a linearized classical system, and brings
nonlinear terms into a corresponding normal form. The obtained expressions
reproduce the exact low-energy spectrum of the system remarkably well even for
a small number of particles N corresponding to fillings of just two particles
per site. Such small fillings are often used in current experiments, and it is
inspiring to get insight into this quantum regime using essentially classical
calculations.
