• DOI:10.1007/s00220-022-04575-8
• Corpus ID: 251564371

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 potentials X j and Y j are Floquet isospectral, j = 1 , 2. This allows us to prove that for any non-constant separable real-valued Γ-periodic potential, the Fermi variety F λ ( V ) / Z 2 is irreducible for any λ ∈ C , which partially confirms a conjecture of Gieseker, Kn¨orrer and Trubowitz in the early 1990s.