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  1. Free, publicly-accessible full text available June 12, 2025
  2. 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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    Free, publicly-accessible full text available July 1, 2025
  3. Free, publicly-accessible full text available February 1, 2025
  4. 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.

     
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    Free, publicly-accessible full text available January 1, 2025
  5. 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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  6. We show that the sublinear bound of the bad Green’s functions implies explicit logarithmic bounds of moments for long range operators in arbitrary dimension.

     
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  7. 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.

     
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