Weak separability and partial Fermi isospectrality of discrete periodic Schrödinger operators
Jifeng Chu, Kang Lyu, Chuan-Fu Yang
TL;DR
The paper addresses inverse spectral questions for discrete periodic Schrödinger operators $Δ+V$ on $\mathbb{Z}^d$ with coprime periodicities, introducing generalized partial Fermi isospectrality and weak separability to capture partial spectral information. It develops a framework showing that generalized partial Fermi isospectrality enforces equality of weak separability, enabling recovery of $(d_1,\dots,d_r)$-separability from partial, rather than full, isospectral data, and reveals that components of generalized Fermi isospectral potentials are Floquet isospectral up to constants in a precise sense. The results extend prior work by replacing full Fermi/Floquet isospectrality with projections of Bloch/Fermi data onto lower-dimensional subspaces, making separability conclusions applicable under weaker hypotheses. An Ambarzumian-type rigidity result further asserts that if a potential is generalized partially Fermi isospectral to a constant, then the potential must be constant, highlighting strong structural constraints in the discrete periodic setting.
Abstract
In this paper, we consider the discrete periodic Schrödinger operators $Δ+V$ on $\Z^d$, where $V$ is $Γ$-periodic with $Γ=q_1 \mathbb{Z}\oplus q_2\mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ and positive integers $q_j$, $j=1,2,\cdots,d,$ are pairwise coprime. We introduce the notions of generalized partial Fermi isospectrality and weak separability, and prove that two generalized partially Fermi isospectral potentials have the same weak separability. As a direct application, we can prove that two potentials have the same $(d_1,d_2,\cdots,d_r)$-separability by assuming that they are generalized partially Fermi isospectral, instead of the Fermi isospectrality or Floquet isospectrality. Besides, we prove that each couples of components of the generalized Fermi isospectral potentials are Floquet isospectral in some sense.