Ordering of Energy Levels in the Fröhlich Model

Consider a one-dimensional system of $N$ electrons subject to an external potential $U$. Let $E_{\mathrm{e}\mathrm{l}}(S)$ denote the ground state energy of the system with total spin $S$. The Mattis–Lieb theorem asserts that, for a broad class of potentials $U$, the inequality $E_{\mathrm{e}\mathrm{l}}(S)<E_{\mathrm{e}\mathrm{l}}(S^{\mathrm{\prime }})$ holds whenever $S<S^{\mathrm{\prime }}$. This result implies that the ground state of a one-dimensional many-electron system is non-ferromagnetic. In the present work, we demonstrate that the Mattis–Lieb theorem can be extended to electron-phonon interacting systems governed by the Fröhlich model. Our analysis is carried out in the setting without an ultraviolet cutoff. The cornerstone of our approach is the construction of a Feynman–Kac-type formula for the heat semigroup generated by the Fröhlich Hamiltonian.