Monotonicity of the polaron energy

In condensed matter physics, the polaron has been fascinating subject. It is described by the Hamiltonian of H. Fr\"ohlich. In this paper, the Fr\"ohlich Hamiltonian is investigated from a viewpoint of the self-dual cone analysis proposed in Miyao (2011). This point of view clarifies the monotonicity of polaron energy, i.e., denoting the lowest energy of the Fr\"ohlich Hamiltonian with the ultraviolet cutoff $\Lambda$ by $E_{\Lambda}$, we prove $E_{\Lambda}>E_{\Lambda'}$ for $\Lambda<\Lambda'$.