Fermi isospectrality for discrete periodic Schrödinger operators

• Published in: Communications on pure and applied mathematics Vol. 77; no. 2; pp. 1126 - 1146
• Main Author: Liu, Wencai
• Format: Journal Article
• Language: English
• Published: New York John Wiley and Sons, Limited 01.02.2024
• Subjects: Energy levels; Operators (mathematics)
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.22161

Let Γ=q1Z⊕q2Z⊕…⊕qdZ$\Gamma =q_1\mathbb {Z}\oplus q_2 \mathbb {Z}\oplus \ldots \oplus q_d\mathbb {Z}$, where ql∈Z+$q_l\in \mathbb {Z}_+$, l=1,2,…,d$l=1,2,\ldots ,d$, are pairwise coprime. Let Δ+V$\Delta +V$ be the discrete Schrödinger operator, where Δ is the discrete Laplacian on Zd$\mathbb {Z}^d$ and the potential V:Zd→C$V:\mathbb {Z}^d\rightarrow \mathbb {C}$ is Γ‐periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension d≥3$d\ge 3$: (1)If at some energy level, Fermi varieties of two real‐valued Γ‐periodic potentials V and Y are the same (this feature is referred to as Fermi isospectrality of V and Y), and Y is a separable function, then V is separable; (2)If two complex‐valued Γ‐periodic potentials V and Y are Fermi isospectral and both V=⨁j=1rVj$V=\bigoplus _{j=1}^rV_j$ and Y=⨁j=1rYj$Y=\bigoplus _{j=1}^r Y_j$ are separable functions, then, up to a constant, lower dimensional decompositions Vj$V_j$ and Yj$Y_j$ are Floquet isospectral, j=1,2,…,r$j=1,2,\ldots ,r$; (3)If a real‐valued Γ‐potential V and the zero potential are Fermi isospectral, then V is zero. In particular, all conclusions in (1), (2) and (3) hold if we replace the assumption “Fermi isospectrality” with a stronger assumption “Floquet isospectrality”.