| Abstract: |
| We consider the discrete periodic Schr\odinger operators $\Delta+V$ on $\Z^d$, where $V$ is $\Gamma$-periodic with $\Gamma=q_1 \mathbb{Z}\oplus q_2\mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ and positive integers $q_j$, $j=1,2,\cdots,d$ are pairwise coprime. By introducing the notions of generalized partial Fermi isospectrality and weak separability, we prove that two generalized partially Fermi isospectral potentials have the same weak separability. As a direct application, we prove that two potentials have the same $(d_1,d_2,\cdots,d_r)$-separability by assuming that the projections of Fermi varieties or Bloch varieties on some $3$-dimensional subspace are the same, instead of the coincidence of the whole Fermi varieties or Bloch varieties. Besides, we prove that each couples of components of the generalized Fermi isospectral potentials are Floquet isospectral. |
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