# Fermi isospectrality for discrete periodic Schrodinger operators

@inproceedings{Liu2021FermiIF, title={Fermi isospectrality for discrete periodic Schrodinger operators}, author={Wencai Liu}, year={2021} }

. Let Γ = q 1 Z ⊕ q 2 Z ⊕· · ·⊕ q d Z , where q l ∈ Z + , l = 1 , 2 , · · · , d . Let ∆+ V be the discrete Schr¨odinger operator, where ∆ is the discrete Laplacian on Z d and the potential V : Z d → R is Γ-periodic. We prove three rigidity theorems for discrete periodic Schr¨odinger operators in any dimension d ≥ 3: In particular, all conclusions in (1), (2) and (3) hold if we replace the assumption “Fermi isospectrality” with a stronger assumption “Floquet isospectrality”.

## 10 Citations

### Algebraic Properties of the Fermi Variety for Periodic Graph Operators

- Mathematics, Computer Science
- 2023

It is shown how the abstract bound implies irreducibility in many lattices of interest, including examples with more than one vertex in the fundamental cell such as the Lieb lattice as well as certain models obtained by the process of graph decoration.

### Bloch varieties and quantum ergodicity for periodic graph operators

- Mathematics
- 2022

. For periodic graph operators, we establish criteria to determine the overlaps of spectral band functions based on Bloch varieties. One criterion states that for a large family of periodic graph…

### Critical points of discrete periodic operators

- Mathematics
- 2020

We study the spectrum of operators on periodic graphs using methods from combinatorial algebraic geometry. Our main result is a bound on the number of complex critical points of the Bloch variety,…

### Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues

- MathematicsGeometric and Functional Analysis
- 2022

Let $$H_0$$ H 0 be a discrete periodic Schrödinger operator on $$\ell ^2(\mathbb {Z}^d)$$ ℓ 2 ( Z d ) : $$\begin{aligned} H_0=-\Delta +V, \end{aligned}$$ H 0 = - Δ + V , where $$\Delta $$ Δ is the…

### Fermi Isospectrality of Discrete Periodic Schrödinger Operators with Separable Potentials on $$\mathbb {Z}^2$$ Z 2

- MathematicsCommunications in Mathematical Physics
- 2022

Given two coprime numbers $$q_1$$ q 1 and $$q_2$$ q 2 , let $$\Gamma =q_1\mathbb {Z}\oplus q_2 \mathbb {Z} $$ Γ = q 1 Z ⊕ q 2 Z . Let $$\Delta +X$$ Δ + X be the discrete periodic Schrödinger operator…

### Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators

- MathematicsJournal of Functional Analysis
- 2022

### Floquet isospectrality for periodic graph operators

- Mathematics
- 2023

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ with arbitrary positive integers $q_l$, $l=1,2,\cdots,d$. Let $\Delta_{\rm discrete}+V$ be the discrete Schr\"odinger…

### Topics on Fermi varieties of discrete periodic Schrödinger operators

- MathematicsJournal of Mathematical Physics
- 2022

This is a survey of recent progress on the irreducibility of Fermi varieties, rigidity results and embedded eigenvalue problems of discrete periodic Schrödinger operators.

### Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues

- Materials ScienceGeometric and Functional Analysis
- 2022

Let H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}…

### Fermi Isospectrality of Discrete Periodic Schrödinger Operators with Separable Potentials on Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{

- Communications in Mathematical Physics
- 2022

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