arXiv subjects

Zhiwen Duan

Publications and source records attributed to Zhiwen Duan.

13 recordsLinked to original sources

arXiv2026

Observability Inequalities and the Logvinenko--Sereda Theorem for the Dunkl Transform

Let $R$ be a normalized root system in $\mathbb{R}^d$ with reflection group $G$, let $k$ be a $G$-invariant multiplicity function, and let $\mathcal{F}_k$ be the associated Dunkl transform. We write $k_α=k(α)>0$ and $\mathrm{d}μ_k(x)=w(x)\,\mathrm{d}x$, where $w(x)=\prod_{α\in R_+}|\langle x,α\rangle|^{2k_α}$. We study observability, Hölder-type interpolation, and spectral inequalities for the Dunkl heat equation on $\mathbb{R}^d$. We establish a Bernstein inequality for ordinary derivatives and a Logvinenko--Sereda theorem with a spectral constant of the form $e^{C(1+N)}$ for functions whose Dunkl transforms are supported in $\overline{B(0,N)}$. We characterize observable sets as the measurable sets that are thick with respect to $μ_k$, and prove the equivalence of the observability, Hölder-type interpolation, and spectral inequalities.

math.AP↗

arXiv2026

Double-logarithmic stability for a parabolic inverse problem on CTA manifolds

We establish a double-logarithmic stability estimate for the recovery of a time-dependent potential in the parabolic equation $(c^{-1}\partial_t-Δ_g+q)u=0$ on a conformally transversally anisotropic manifold. The partial input--output map associates an initial state and lateral Dirichlet data supported in a prescribed open set with the terminal state and the Neumann trace on another open set. These sets contain the positive and negative boundary parts determined by a limiting Carleman weight, respectively. The measurement discrepancy is the operator norm of the difference of two such maps, whose range has improved Sobolev regularity. We prove stability under a uniform stability assumption for the attenuated geodesic ray transform on a family of non-tangential geodesics. This assumption is verified when the transversal manifold is simple and for regular families on certain non-simple manifolds. The proof uses Gaussian beam quasimodes with a uniform $O(h^{1/2})$ concentration estimate, geometric optics solutions obtained from boundary Carleman estimates, and stability of the geodesic ray transform for small attenuation. Quantitative analytic continuation for Hilbert-space-valued Fourier transforms and Sobolev interpolation yield the double-logarithmic modulus.

math.AP↗

arXiv2026

Observability and controllability for the Schrödinger equation on polyhedra

We consider the observability and controllability for the Schrödinger equation on polyhedra. We show that for the Schrödinger equation on convex polyhedra in $\mathbb{R}^d\,(d\geq2)$, when the initial data possess $H^s\,(s>d/2)$ regularity, observability and controllability hold, with the control region being an arbitrary nonempty open neighborhood of the singular set of the boundary, which is the same as the control region for eigenfunctions. Our proof of observability is a concrete adaptation of the method in the Burq--Zworski black box control work, where the observability inequality is proved via a resolvent estimate. Hence, before proving observability, we present an observability resolvent estimate on polyhedra. The idea of the proof of this resolvent estimate comes from the case of eigenfunction concentration on polyhedra considered by Cekić--Georgiev--Mukherjee, which uses semiclassical measures and relies closely on the dynamical properties of the billiard flow in polyhedra. Finally, by a standard HUM argument, we show that observability implies controllability.

math.OC↗

arXiv2026

Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform

We establish quantitative uniqueness and a fractal uncertainty principle for the Fourier Bessel transform. In arbitrary dimension, we prove a quantitative uniqueness estimate on relatively dense sets for functions whose Fourier Bessel transforms decay according to a quasi-analytic weight. In dimension one, if $X\subset[0,1]$ and $Y\subset[a,a+h^{-1}]$ are$δ$-regular on the relevant scales, then, for $a\geq a_0h^{-1}$, \[ \operatorname{supp}\mathcal H_νf\subset Y \quad\Longrightarrow\quad \|\mathbf 1_Xf\|_{L^2_ν} \leq Ch^β\|f\|_{L^2_ν}. \] The lack of translation invariance prevents a direct application of the classical Fourier argument. We overcome this by constructing damping functions adapted to translated regular sets and combining Beurling Malliavin multipliers with large-argument Bessel asymptotics and a Bourgain Dyatlov multiscale iteration.

math.CA↗

arXiv2026

Time-Dependent Potential Recovery from Local Data on Conformally Transversally Anisotropic Manifolds

We study the recovery of a time-dependent potential in a parabolic equation on a conformally transversally anisotropic manifold. The initial state and a lateral Dirichlet datum supported near the front face are used as inputs, while the terminal state and the Neumann trace near the back face are observed. Assuming injectivity of the attenuated geodesic ray transform on the transversal manifold for all sufficiently small constant attenuations, we prove that these partial input-output data uniquely determine the potential. No simplicity assumption is imposed on the transversal manifold. The proof combines a conformal reduction, transversal Gaussian beam quasimodes, boundary Carleman estimates, and geometric optics solutions with prescribed lateral supports.

math.AP↗

arXiv2026

Unique continuation inequalities for the Dunkl-Schrödinger equation via uncertainty principles

In this paper, we establish unique continuation inequalities at two time points for the Dunkl--Schrödinger equation. The proof is based on quantitative uncertainty principles for the Dunkl transform. In particular, we prove that pairs of (\varepsilon,k)-thin sets form strong annihilating pairs for the Dunkl transform, which yields quantitative unique continuation properties for solutions to the Dunkl--Schrödinger equation.

math.AP↗

arXiv2025

Almost everywhere convergence of Bochner-Riesz means for the Hermite type Laguerre expansions

Consider the space $\mathbb{R}_+^d=(0,\infty)^d$ equipped with Euclidean distance and the Lebesgue measure. For every $α=(α_1,...,α_d)\in[-1/2,\infty)^d$, we consider the Hermite-Laguerre operator $\mathcal{L}^α=-Δ+\arrowvert x\arrowvert^2+\sum_{i=1}^{d}(α_j^2-\frac{1}{4})\frac{1}{x_i^2}$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $\mathcal{L}^α$ which is defined as $S_R^λ(\mathcal{L}^α)f(x)=\sum_{n=0}^{\infty}(1-\frac{4n+2\arrowvertα\arrowvert_1+2d}{R^2})_{+}^λ\mathcal{P}_nf(x)$. Here $\mathcal{P}_nf(x)$ is the n-th Laguerre spectral projection operator and $\arrowvertα\arrowvert_1$ denotes $\sum_{i=1}^{d}α_i$. For $2\leq p<\infty$, we prove that \[ \lim_{R \to \infty} S_R^λ(\mathcal{L}^α)f = f \quad \text{a.e.} \] for all $f\in L^p({\mathbb{R}_+^d})$ provided that $λ>λ(p)/2$ and $λ(p)=\max\{d(1/2-1/p)-1/2,0\}$. Conversely, we show that the convergence generally fails if $λ<λ(p)/2$ in the sense that there exists $f\in L^p({\mathbb{R}_+^d})$ for $2d/(d-1)< p$ such that the convergence fails.

math.FA↗

arXiv2025

Uncertainty Principle and Geometric Condition for the Observability of Schrödinger Equations

We provide necessary and sufficient geometric conditions for the exact observability of the Schrödinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for generalized Fourier transform. Specifically, the generalized Fourier transform associated with the Schrödinger operator with inverse-square potentials on the half-line is the well-known Hankel transform. We present a necessary and sufficient condition for a subset $Ω$, such that a function whose Hankel transform is supported in a given interval can be bounded, in the $L^2$-norm, from above by its restriction to $Ω$, with a constant independent of the position of the interval.

math.AP↗

arXiv2024

A 2D T-carbon 2-(111) structure with tunable electric and optical properties via chemical decorations: a first-principles investigation

We proposed a new two-dimensional carbon material named 2-(111) planar T-carbon, which is obtained by slicing bulk T-carbon along its (111) crystallographic direction. 2-(111) planar T-carbon's optical and electrical properties can be engineered via surface decoration. Comparing the DFT phonon spectra of pristine and five decorated 2-(111) planar T-carbon obtained by first-principles calculations, we conclude that surface decoration presents a promising, effective, and feasible strategy to improve the structural stability of 2-(111) planar T-carbon. The calculated band structures and electronic properties show direct electronic band gap values between 0.17 eV (-O= decorated) and 2.21 eV (Hydrogenated). Chemical decoration also promises blue or red energy shifts in its optical properties.

cond-mat.mtrl-sci↗

arXiv2023

Observability and unique continuation inequalities for the Schrödinger equations with inverse-square potentials

This paper is inspired by Wang, Wang and Zhang's work [ Observability and unique continuation inequalities for the Schrödinger equation. J. Eur. Math. Soc. 21, 3513--3572 (2019)], where they present several observability and unique continuation inequalities for the free Schrödinger equation in $\mathbb{R}^{n}$. We extend all such observability and unique continuation inequalities for the Schrödinger equations on half-line with inverse-square potentials. Technically, the proofs essentially rely on the representation of the solution, a Nazarov type uncertainty principle for the Hankel transform and an interpolation inequality for functions whose Hankel transform have compact support.

math.AP↗

arXiv2023

Scattering for the fractional magnetic Schrodinger operators

In this paper, we prove the existence of the scattering operator for the fractional magnetic Schrodinger operators. For this, we construct the fractional distorted Fourier transforms with magnetic potentials. Applying the properties of the distorted Fourier transforms, the existence and the asymptotic completeness of the wave operators are obtained. Furthermore, we prove the absence of positive eigenvalues for Schrodinger operators.

math.AP↗

arXiv2023

Dispersive decay estimates for the magnetic Schrödinger equations

In this paper, we present a proof of dispersive decay for both linear and nonlinear magnetic Schrödinger equations. To achieve this, we introduce the fractional distorted Fourier transforms with magnetic potentials and define the fractional differential operator $\arrowvert J_{A}(t)\arrowvert^{s}$. By leveraging the properties of the distorted Fourier transforms and the Strichartz estimates of $\arrowvert J_{A}\arrowvert^{s}u$, we establish the dispersive bounds with the decay rate $t^{-\frac{n}{2}}$. This decay rate provides valuable insights into the spreading properties and long-term dynamics of the solutions to the magnetic Schrödinger equations.

math.AP↗

arXiv2018

$L^p$ estimates for fractional schrodinger operators with kato class potentials

Let $α>0$, $H=(-\triangle)^α+V(x)$, $V(x)$ belongs to the higher order Kato class $K_{2α}(\mathbbm{R}^n)$. For $1\leq p\leq \infty$, we prove a polynomial upper bound of $\|e^{-itH}(H+M)^{-β}\|_{L^p, L^p}$ in terms of time $t$. Both the smoothing exponent $β$ and the growth order in $t$ are almost optimal compared to the free case. The main ingredients in our proof are pointwise heat kernel estimates for the semigroup $e^{-tH}$. We obtain a Gaussian upper bound with sharp coefficient for integral $α$ and a polynomial decay for fractal $α$.

math.AP↗