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  1. ; ; (, Nonlinearity)
    Abstract This paper investigates uniqueness results for perturbed periodic Schrödinger operators on Z d . Specifically, we consider operators of the form H = − Δ + V + v , where Δ is the discrete Laplacian, V : Z d → R is a periodic potential, and v : Z d → C represents a decaying impurity. We establish quantitative conditions under which the equation − Δ u + V u + v u = λ u , for λ ∈ C , admits only the trivial solution u ≡ 0 . Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes ofλ. 
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    Free, publicly-accessible full text available April 9, 2026
  2. (, International Mathematics Research Notices)
    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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  3. (, Communications on Pure and Applied Mathematics)
    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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  4. ; (, SIAM Journal on Applied Algebra and Geometry)
    Free, publicly-accessible full text available March 31, 2026
  5. ; (, Communications in Mathematical Physics)
    Free, publicly-accessible full text available November 1, 2025
  6. (, Journal d'Analyse Mathématique)
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  7. ; ; ; ; ; ; ; (, Journal of Mathematical Physics)
    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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  8. ; (, Journal of the European Mathematical Society)
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  9. ; (, Journal of Spectral Theory)
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  10. ; ; (, Journal of Functional Analysis)
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