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}
}
  • Wencai Liu
  • Published 1 November 2021
  • Mathematics
  • Journal of Mathematical Physics
This is a survey of recent progress on the irreducibility of Fermi varieties, rigidity results and embedded eigenvalue problems of discrete periodic Schrödinger operators. 
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11 Citations
Background Citations
4

11 Citations

Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators

Algebraic Properties of the Fermi Variety for Periodic Graph Operators

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

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

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

. 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

. 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

  • Wencai Liu
  • Mathematics
    Geometric 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

  • Wencai Liu
  • Mathematics
    Communications 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

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

  • Wencai Liu
  • Materials Science
    Geometric and Functional Analysis
  • 2022
Let H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}…

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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…

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Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators

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

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

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