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

@article{Liu2022FermiIO,
  title={Fermi Isospectrality of Discrete Periodic Schr{\"o}dinger Operators with Separable Potentials on \$\$\mathbb \{Z\}^2\$\$},
  author={Wencai Liu},
  journal={Communications in Mathematical Physics},
  year={2022}
}
  • Wencai Liu
  • Published 15 August 2022
  • Mathematics
  • Communications in Mathematical Physics
. Let Γ = q 1 Z ⊕ q 2 Z with q 1 ∈ Z + and q 2 ∈ Z + . Let ∆+ X be the discrete periodic Schr¨odinger operator on Z 2 , where ∆ is the discrete Laplacian and X : Z 2 → C is Γ-periodic. In this paper, we develop tools from complex analysis to study the isospectrality of discrete periodic Schr¨odinger operators. We prove that if two Γ-periodic potentials X and Y are Fermi isospectral and both X = X 1 ⊕ X 2 and Y = Y 1 ⊕ Y 2 are separable functions, then, up to a constant, one dimensional… 

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

Topics on Fermi varieties of discrete periodic Schrödinger operators

  • Wencai Liu
  • Mathematics
    Journal 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.

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

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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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