# Topics on Fermi varieties of discrete periodic Schrödinger operators

@article{Liu2021TopicsOF, title={Topics on Fermi varieties of discrete periodic Schr{\"o}dinger operators}, author={Wencai Liu}, journal={Journal of Mathematical Physics}, year={2021} }

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

## 11 Citations

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

- MathematicsJournal of Functional Analysis
- 2022

### 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.

### Analytic and algebraic properties of dispersion relations (Bloch varieties) and Fermi surfaces. What is known and unknown

- Mathematics
- 2023

The article surveys the known results and conjectures about the analytic properties of dispersion relations and Fermi surfaces for periodic equations of mathematical physics and their spectral…

### Irreducibility of the Dispersion Relation for Periodic Graphs

- Mathematics
- 2023

Recent work of Liu investigated the irreducibility of Fermi varieties for the Grid graph, and work of Fillman, Liu and Matos investigated the irreducibility of Bloch varieties for a wide class of…

### 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…

### Fermi isospectrality for discrete periodic Schrodinger operators

- Mathematics
- 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 →…

### 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…

### 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…

### 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}…

## 63 References

### A Rellich type theorem for discrete Schrödinger operators

- Mathematics
- 2012

An analogue of Rellich's theorem is proved for discrete Laplacians on square lattices, and applied to show unique continuation properties on certain domains as well as non-existence of embedded…

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

- MathematicsJournal of Functional Analysis
- 2022

### On isospectral periodic potentials on a discrete lattice I

- Mathematics
- 1988

On considere une version discretisee et generalisee de l'equation de Schrodinger et on etudie le probleme aux valeurs propres

### Inverse spectral problem for the Schrödinger equation with periodic vector potential

- Mathematics
- 1989

For the Schrödinger operator with periodic magnetic (vector) and electric (scalar) potentials a new system of spectral invariants is found. These invariants are enough to prove the rigidity of…

### Periodic and limit-periodic discrete Schrödinger operators

- Mathematics
- 2011

The theory of discrete periodic and limit-periodic Schr\"odinger operators is developed. In particular, the Floquet--Bloch decomposition is discussed. Furthermore, it is shown that an arbitrarily…

### Examples of fourth-order scattering-type operators with embedded eigenvalues in their continuous spectra

- Mathematics
- 2021

We give examples of fourth-order scattering-type operators, acting on L2(R), which have eigenvalues embedded in their continuous spectra.

### Discrete Bethe–Sommerfeld Conjecture

- Mathematics
- 2017

In this paper, we prove a discrete version of the Bethe–Sommerfeld conjecture. Namely, we show that the spectra of multi-dimensional discrete periodic Schrödinger operators on $${\mathbb{Z}^d}$$Zd…

### An overview of periodic elliptic operators

- Mathematics
- 2015

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic…

### Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach

- Mathematics
- 2018

ABSTRACT We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric…