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Free, publicly-accessible full text available June 12, 2025
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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.
Free, publicly-accessible full text available July 1, 2025 -
Free, publicly-accessible full text available February 1, 2025
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We consider a discrete non-linear Schrödinger equation on Z and show that, after adding a small potential localized in the time-frequency space, one can construct a three-parametric family of non-decaying spacetime quasiperiodic solutions to this equation. The proof is based on the Craig–Wayne–Bourgain method combined with recent techniques of dealing with Anderson localization for two-dimensional quasiperiodic operators with degenerate frequencies.
Free, publicly-accessible full text available January 1, 2025 -
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 potentials
V andY are the same (this feature is referred to asFermi isospectrality ofV andY ), andY is a separable function, thenV is separable;If two complex‐valued Γ‐periodic potentials
V andY are 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‐potential
V and the zero potential are Fermi isospectral, thenV is zero. -
Abstract We discover that the distribution of (frequency and phase) resonances plays a role in determining the spectral type of supercritical quasi-periodic Schrödinger operators. In particular, we disprove the 2nd spectral transition line conjecture of Jitomirskaya in the early 1990s.