[Paper Review] Fermi isospectrality for discrete periodic Schrodinger operators

Wencai Liu|arXiv (Cornell University)|Jun 7, 2021

Spectral Theory in Mathematical Physics24 references4 citations

TL;DR

This paper establishes rigidity theorems for discrete periodic Schrödinger operators on $bZ^d$, $d \geq 3$, proving that Fermi isospectrality—equality of Fermi varieties at a fixed energy—implies structural constraints on the potential: if the potential $V$ is Fermi isospectral to a separable potential $Y$, then $V$ must also be separable; if both $V$ and $Y$ are separable, their components are Floquet isospectral; and if $V$ is Fermi isospectral to the zero potential, then $V$ must be identically zero. These results resolve inverse spectral questions in the discrete setting.

ABSTRACT

Let $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, where $q_l\in \mathbb{Z}_+$, $l=1,2,\cdots,d$. Let $Δ+V$ be the discrete Schrödinger operator, where $Δ$ is the discrete Laplacian on $\mathbb{Z}^d$ and the potential $V:\mathbb{Z}^d o \mathbb{R}$ is $Γ$-periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension $d\geq 3$: (1) if at some energy level, Fermi varieties of the $Γ$-periodic potential $V$ and the $Γ$-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 $V=\bigoplus_{j=1}^rV_j$ and $Y=\bigoplus_{j=1}^r Y_j$ 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".

Motivation & Objective

• To resolve inverse spectral problems for discrete periodic Schrödinger operators in dimensions $d \geq 3$ using Fermi isospectrality.
• To determine whether Fermi isospectrality of two potentials implies structural equivalence, such as separability or component-wise Floquet isospectrality.
• To prove that Fermi isospectrality with the zero potential forces the potential to be identically zero.
• To extend known rigidity results from the continuous to the discrete setting, particularly for separable and completely separable potentials.
• To establish that Floquet isospectrality implies Fermi isospectrality, and that the stronger condition yields the same rigidity conclusions.

Proposed method

• Define the discrete Schrödinger operator $H_0 = \Delta + V$ on $\ell^2(\bbZ^d)$, where $V$ is $\Gamma$-periodic with $\Gamma = q_1\bbZ \oplus \cdots \oplus q_d\bbZ$, $q_l \in \bbZ_+$.
• Use the Floquet-Bloch decomposition to define the operator $D_V(k)$ with quasi-periodic boundary conditions, and define the Fermi variety $F_\lambda(V)$ as the set of $k \in \bbC^d$ for which $\lambda$ is an eigenvalue of $D_V(k)$.
• Introduce the concept of Fermi isospectrality: $F_{\lambda_0}(V) = F_{\lambda_0}(Y)$ for some $\lambda_0 \in \bbC$.
• Prove that if $V$ and $Y$ are Fermi isospectral and $Y$ is separable, then $V$ must also be separable, using analytic continuation and unique factorization of characteristic polynomials.
• For separable potentials $V = \bigoplus_{j=1}^r V_j$, $Y = \bigoplus_{j=1}^r Y_j$, show that Fermi isospectrality implies that each pair $(V_j, Y_j)$ is Floquet isospectral, via asymptotic analysis of eigenvalue branches and coefficient comparison in characteristic polynomials.
• Use coefficient comparison in the characteristic polynomial $\tilde{\mathcal{P}}_V(z,\lambda)$ to show that if $V$ is Fermi isospectral to the zero potential, then all Fourier coefficients of $V$ vanish, implying $V \equiv 0$.

Experimental results

Research questions

• RQ1 If two $\Gamma$-periodic potentials $V$ and $Y$ are Fermi isospectral and $Y$ is separable, must $V$ also be separable?
• RQ2 If both $V$ and $Y$ are separable and Fermi isospectral, are their lower-dimensional components $V_j$ and $Y_j$ Floquet isospectral?
• RQ3 If a potential $V$ is Fermi isospectral to the zero potential, must $V$ be identically zero?
• RQ4 Does Fermi isospectrality imply the same structural constraints as Floquet isospectrality in the discrete periodic Schrödinger setting?
• RQ5 Can rigidity results for the discrete case be established under weaker assumptions than full Floquet isospectrality?

Key findings

• If $V$ and $Y$ are Fermi isospectral and $Y$ is separable, then $V$ must also be separable, even if $V$ is not assumed to be separable a priori.
• If $V = \bigoplus_{j=1}^r V_j$ and $Y = \bigoplus_{j=1}^r Y_j$ are both separable and Fermi isospectral, then for each $j$, the potentials $V_j$ and $Y_j$ are Floquet isospectral, up to a constant shift.
• If $V$ is Fermi isospectral to the zero potential, then $V \equiv 0$, proving a strong rigidity result in the discrete setting.
• The results hold under the weaker assumption of Fermi isospectrality, and the same conclusions follow if the stronger condition of Floquet isospectrality is assumed.
• The proof relies on asymptotic analysis of eigenvalue branches $\lambda^l(z_d)$ as $|z_d| \to \infty$, showing that the leading-order behavior determines the structure of the potential.
• Coefficient comparison in the characteristic polynomial $\tilde{\mathcal{P}}_V(z,\lambda)$—specifically the coefficients of $\lambda^{Q_1-1}$ and $\lambda^{Q_1-2}$—yields that $V \equiv 0$ when $V$ is Fermi isospectral to the zero potential.