Stability of the Nagaoka state in the one-band Hubbard model

We discuss the stability of the saturated ferromagnetic state of the one-band Hubbard model in the thermodynamic limit. We prove rigorously that the Nagaoka state is stable on a d-dimensional (d=2,3) simple-cubic lattice if the number of holes ${\mathit{N}}_{\mathit{h}}$ is less than ${\mathit{N}}_{\mathrm{\Lambda }}^{\mathrm{\alpha }}$, 0≤α<1/2d. Finally, we explain briefly why the Nagaoka state is probably unstable when ${\mathit{N}}_{\mathit{h}}$≫${\mathit{N}}_{\mathrm{\Lambda }}^{1/2\mathit{d}}$.