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A counterexample to Fermi isospectral rigidity for two dimensional discrete periodic Schrödinger operators

Taylor Brysiewicz, Matthew Faust, Wencai Liu

Abstract

Using numerical certification, we prove the existence of a nontrivial real-valued two dimensional periodic potential whose associated discrete Schrödinger operator is Fermi isospectral to the zero potential. This provides a negative answer to a question posed by the third author concerning the rigidity of Fermi isospectrality in dimension two. This example also disproves a conjecture of Gieseker, Knörrer, and Trubowitz in the 1990s stating that for any nontrivial real-valued periodic potential in dimension two, the Fermi variety is irreducible at all energy levels.

A counterexample to Fermi isospectral rigidity for two dimensional discrete periodic Schrödinger operators

Abstract

Using numerical certification, we prove the existence of a nontrivial real-valued two dimensional periodic potential whose associated discrete Schrödinger operator is Fermi isospectral to the zero potential. This provides a negative answer to a question posed by the third author concerning the rigidity of Fermi isospectrality in dimension two. This example also disproves a conjecture of Gieseker, Knörrer, and Trubowitz in the 1990s stating that for any nontrivial real-valued periodic potential in dimension two, the Fermi variety is irreducible at all energy levels.
Paper Structure (8 sections, 7 theorems, 29 equations)

This paper contains 8 sections, 7 theorems, 29 equations.

Table of Contents

  1. Introduction and main results
  2. Floquet matrices and polynomial formulation
  3. Floquet matrices
  4. Reduction to an algebraic system
  5. Computational proof of Theorem \ref{['thm1']}
  6. Generating the system
  7. Krawczyk's method
  8. Applying Krawczyk's method

Key Result

Theorem 1.1

liu2021fermi Let $d\ge 3$. The only real potential $V$ which is Fermi isospectral to the zero potential $\textbf{0}$ is $\textbf{0}$ itself.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Conjecture 1
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2
  • ...and 1 more