Instanton solutions on the polymer harmonic oscillator

It is computed, using instanton methods, the first allowed energy band for the polymer harmonic oscillator. The result is consistent with the band structure of the standard quantum pendulum but with pure point spectrum. An effective infinite degeneracy emerges in the formal limit $\mu \mathrm{/}{l}_0\to 0$ where $l_0$ is the characteristic length of the vacuum eigenfunction of a quantum harmonic oscillator. As an additional result, it is shown along the article the role played by the lattice reference point $\lambda$ in the full quantization of the polymer harmonic oscillator.