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

  title={Irreducibility of the Fermi variety for discrete periodic Schr{\"o}dinger operators and embedded eigenvalues},
  author={Wencai Liu},
  journal={Geometric and Functional Analysis},
  • Wencai Liu
  • Published 8 June 2020
  • Mathematics
  • Geometric and Functional Analysis
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 discrete Laplacian and $$V:\mathbb {Z}^d\rightarrow \mathbb {C}$$ V : Z d → C is periodic. We prove that for any $$d\ge 3$$ d ≥ 3 , the Fermi variety at every energy level is irreducible (modulo periodicity). For $$d=2$$ d = 2 , we prove that the Fermi variety at every energy level except for the… 
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