Floquet Isospectrality of the Zero Potential for Discrete Periodic Schrödinger Operators

2024/01/18 by Faust, Matthew, Liu, Wencai, Matos, Rodrigo +5 · 1 citation
#35J10 #35P05 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 58J53. Secondary: 47B36 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2401.09731

Abstract

Let Γ=q₁ℤ⊕ q₂ ℤ⊕⋯⊕ q_dℤ, with q_j∈ (ℤ⁺)^d for each j∈ \1,…,d\, and denote by Δ the discrete Laplacian on ℓ²( ℤ^d). Using Macaulay2, we first numerically find complex-valued Γ-periodic potentials V:ℤ^d→ ℂ such that the operators Δ+V and Δ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions.

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• Weak separability and partial Fermi isospectrality of discrete periodic Schrödinger operators

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