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

Monotonicity of the polaron energy