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Creators/Authors contains: "Liu, Wencai"

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  1. ; ; (, Journal of Mathematical Physics)
    We use Bézout’s theorem and Bernstein–Khovanskii–Kushnirenko theorem to analyze the level sets of the extrema of the spectral band functions of discrete periodic Schrödinger operators on Z2. These approaches improve upon previous results of Liu and Filonov–Kachkovskiy. 
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    Free, publicly-accessible full text available June 1, 2026
  2. ; ; (, 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
  3. ; (, Communications in Mathematical Physics)
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  4. (, Journal d'Analyse Mathématique)
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  5. ; (, Journal of the European Mathematical Society)
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  6. ; ; ; ; ; ; ; (, 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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  7. ; ; (, Journal of Functional Analysis)
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  8. (, Journal of Differential Equations)
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  9. ; ; (, Journal of Mathematical Physics)
    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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  10. (, Communications in Mathematical Physics)
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