• Corpus ID: 235358526

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”. 
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10 Citations
Background Citations
3
Methods Citations
1

10 Citations

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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…

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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…

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

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  • 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}{

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29 References

ISOSPECTRAL POTENTIALS ON A DISCRETE LATTICE

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