Wencai Liu

activity

20132026

most citedArithmetic Spectral Transitions for the Maryland Model

36 citations · 389 across the 48 of their papers we have counts for

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15 papers · 1 filter

math.SP2026

Irreducibility of the Bloch variety for periodic Schrödinger operators in arbitrary dimension

Wencai Liu

Let $d \geq 1$ and $V \in C (T^{d}; C)$ . We prove that the Bloch variety of $- Δ + V$ is irreducible modulo periodicity. The proof combines holomorphic continuation of spect…

math.SP2023★ 1 cited

Proof of geometric Borg's Theorem in arbitrary dimensions

Wencai Liu

Let $Δ + V$ be the discrete Schrödinger operator, where $Δ$ is the discrete Laplacian on $Z^{d}$ and potential $V : Z^{d} \to C$ is $Γ$ -periodic with $Γ=q_1\mathb…

math.SP2023★ 1 cited

Algebraic Properties of the Fermi Variety for Periodic Graph Operators

Jake Fillman, Wencai Liu, Rodrigo Matos

We present a method to estimate the number of irreducible components of the Fermi varieties of periodic Schrödinger operators on graphs in terms of suitable asymptotics. Our main t…

math.SP2023

Floquet isospectrality for periodic graph operators

Wencai Liu

Let $Γ = q_{1} Z \oplus q_{2} Z \oplus \dots \oplus q_{d} Z$ with arbitrary positive integers $q_{l}$ , $l = 1, 2, \dots, d$ . Let $Δ_{discrete} + V$ be the discrete Schröd…

math.SP2019★ 3 cited

Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case

Svetlana Jitomirskaya, Wencai Liu

We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic wit…

math.SP2019★ 32 cited

Anderson localization for multi-frequency quasi-periodic operators on $Z^{d}$

Svetlana Jitomirskaya, Wencai Liu, Yunfeng Shi

We establish Anderson localization for general analytic $k$ -frequency quasi-periodic operators on $Z^{d}$ for \textit{arbitrary} $k, d$ .

math.SP2018★ 20 cited

Criteria for embedded eigenvalues for discrete Schrödinger operators

Wencai Liu

In this paper, we consider discrete Schrödinger operators of the form, \begin{equation*} (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n). \end{equation*} We view $H$ as a perturbation of the f…

math.SP2018★ 3 cited

Almost Mathieu operators with completely resonant phases

Wencai Liu

Let $α \in R \ Q$ and $β (α) = lim sup_{n \to \infty} (ln q_{n + 1}) / q_{n} < \infty$ , where $p_{n} / q_{n}$ is the continued fraction approximations to $α$ . Let $(H_…

math.SP2018★ 4 cited

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic Jacobi operators

Wencai Liu, Darren C. Ong

We are interested in diagonal perturbations of a periodic Jacobi operator that introduce embedded eigenvalues in its essential spectrum. Embedding multiple points in the essential…

math.SP2018★ 13 cited

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators

Wencai Liu, Darren C. Ong

In this paper, we consider the Schrödinger equation, \begin{equation*} Hu=-u^{\prime\prime}+(V(x)+V_0(x))u=Eu, \end{equation*} where $V_{0} (x)$ is 1-periodic and $V (x)$ is a decayin…

math.SP2017★ 18 cited

Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies

Wencai Liu, Yunfeng Shi

We prove that the size of the spectral gaps of weakly coupled quasi-periodic Schrödinger operators with Liouville frequencies decays exponentially. As an application, we obtain the…

math.SP2013

Absolutely Continuous Spectrum for the Quasi-periodic Schrödinger Operator in Exponential Regime

Wencai Liu, Xiaoping Yuan

Avila and Jitomirskaya prove that the quasi-periodic Schrödinger operator $H_{λ v, α, θ}$ has purely absolutely continuous spectrum for $α$ in sub-exponential regime (i.e., $β (α) = 0$ )…

math.SP2013★ 10 cited

Anderson localization for the completely resonant phases

Wencai Liu, Xiaoping Yuan

For the almost Mathieu operator $(H_{λ, α, θ} u) (n) = u (n + 1) + u (n - 1) + λ v (θ + n α) u (n)$ , Avila and Jitomirskaya guess that for every phase $ θ\in \mathscr{R} \triangleq\{θ\in \mathbb{R}\;|…

math.SP2013★ 15 cited

Spectral Gaps of Almost Mathieu Operator in Exponential Regime

Wencai Liu, Xiaoping Yuan

For almost Mathieu operator $(H_{λ, α, θ} u)_{n} = u_{n + 1} + u_{n - 1} + 2 λ cos 2 π (θ + n α) u_{n}$ , the dry version of Ten Martini problem predicts that the spectrum $Σ_{λ, α}$ of $H_{λ, α, θ}$ has all…

math.SP2013★ 8 cited

Anderson Localization for the Almost Mathieu Operator in Exponential Regime

Wencai Liu, Xiaoping Yuan

For the almost Mathieu operator $(H_{λ, α, θ} u)_{n} = u_{n + 1} + u_{n - 1} + 2 λ cos 2 π (θ + n α) u_{n}$ , Avila and Jitomirskaya guess that for a.e. $θ$ , $H_{λ, α, θ}$ satisfies Anderson localization if $…

Irreducibility of the Bloch variety for periodic Schrödinger operators in arbitrary dimension

Proof of geometric Borg's Theorem in arbitrary dimensions

Algebraic Properties of the Fermi Variety for Periodic Graph Operators

Floquet isospectrality for periodic graph operators

Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case

Anderson localization for multi-frequency quasi-periodic operators on Zd

Criteria for embedded eigenvalues for discrete Schrödinger operators

Almost Mathieu operators with completely resonant phases

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic Jacobi operators

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators

Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies

Absolutely Continuous Spectrum for the Quasi-periodic Schrödinger Operator in Exponential Regime

Anderson localization for the completely resonant phases

Spectral Gaps of Almost Mathieu Operator in Exponential Regime

Anderson Localization for the Almost Mathieu Operator in Exponential Regime

Anderson localization for multi-frequency quasi-periodic operators on $Z^{d}$