  We discuss the detailed structure of the spectrum of the Hamiltonian for the
polymerized harmonic oscillator and compare it with the spectrum in the
standard quantization. As we will see the non-separability of the Hilbert space
implies that the point spectrum consists of bands similar to the ones appearing
in the treatment of periodic potentials. This feature of the spectrum of the
polymeric harmonic oscillator may be relevant for the discussion of the polymer
quantization of the scalar field and may have interesting consequences for the
statistical mechanics of these models.
