Fermi isospectrality for discrete periodic Schrodinger operators

Abstract

Let $\Gamma =q_1\mathbb{Z}\oplus q_2\mathbb{Z}\oplus \cdots \oplus q_d\mathbb{Z}$, where $q_l\in {\mathbb{Z}}_{+}$, $l=1,2,\cdots ,d$. Let $\Delta +V$ be the discrete Schrödinger operator, where $\Delta$ is the discrete Laplacian on ${\mathbb{Z}}^d$ and the potential $V:{\mathbb{Z}}^d\to \mathbb{R}$ is $\Gamma$-periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension $d\ge 3$: (1) if at some energy level, Fermi varieties of the $\Gamma$-periodic potential $V$ and the $\Gamma$-periodic potential $Y$ are the same (this feature is referred to as {\it Fermi isospectrality} of $V$ and $Y$), and $Y$ is a separable function, then $V$ is separable; (2) if potentials $V$ and $Y$ are Fermi isospectral and both $V={⨁}_{j=1}^r{V}_j$ and $Y={⨁}_{j=1}^r{Y}_j$ are separable functions, then, up to a constant, lower dimensional decompositions $V_j$ and $Y_j$ are Floquet isospectral, $j=1,2,\cdots ,r$; (3) if a 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".

Comm. Pure Appl. Math. to appear

References in corpus (2)

• Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues
• Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators

Cited by in corpus (3)

• Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues
• Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators
• Topics on Fermi varieties of discrete periodic Schrödinger operators