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Let Γ=q1Z⊕q2Z⊕⋯⊕qdZ, with qj∈Z+ for each j ∈ {1, …, d}, and denote by Δ the discrete Laplacian on ℓ2Zd. Using Macaulay2, we first numerically find complex-valued Γ-periodic potentials V:Zd→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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Fermi isospectrality for discrete periodic Schrödinger operators
Abstract Let , where , , are pairwise coprime. Let be the discrete Schrödinger operator, where Δ is the discrete Laplacian on and the potential is Γ‐periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension :If at some energy level, Fermi varieties of two real‐valued Γ‐periodic potentialsVandYare the same (this feature is referred to asFermi isospectralityofVandY), andYis a separable function, thenVis separable;If two complex‐valued Γ‐periodic potentialsVandYare Fermi isospectral and both and are separable functions, then, up to a constant, lower dimensional decompositions and are Floquet isospectral, ;If a real‐valued Γ‐potentialVand the zero potential are Fermi isospectral, thenVis zero.In particular, all conclusions in (1), (2) and (3) hold if we replace the assumption “Fermi isospectrality” with a stronger assumption “Floquet isospectrality”.
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- PAR ID:
- 10478669
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- Volume:
- 77
- Issue:
- 2
- ISSN:
- 0010-3640
- Format(s):
- Medium: X Size: p. 1126-1146
- Size(s):
- p. 1126-1146
- Sponsoring Org:
- National Science Foundation
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