skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


  • Home
  • Search Results
  • Page 1 of 4

Search for: All records

Creators/Authors contains: "Liu, Wencai"

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

  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. 
    more » « less
    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. 
    more » « less
    Free, publicly-accessible full text available April 9, 2026
  3. ; (, Communications in Mathematical Physics)
    Free, publicly-accessible full text available November 1, 2025
  4. (, Journal d'Analyse Mathématique)
    Full Text Available 
  5. ; (, Journal of the European Mathematical Society)
    Full Text Available 
  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. 
    more » « less
    Full Text Available 
  7. ; ; (, Journal of Functional Analysis)
    Full Text Available 
  8. (, Journal of Differential Equations)
    Full Text Available 
  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. 
    more » « less
    Full Text Available 
  10. (, 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”. 
    more » « less