arXiv subjects

Shanlin Huang

Publications and source records attributed to Shanlin Huang.

At least 19 recordsLinked to original sources

arXiv2026

Fractal Remez inequality on the sphere and observability of the heat equation

This paper is concerned with Remez-type inequalities and their applications in observability inequality. Our aim is twofold. First, we establish the following fractal Remez's inequality on the unit sphere $\mathbb{S}^{n-1}$ \begin{align*} \sup_{\mathbb{S}^{n-1}} |p|\le C(M,N,n,\delta)\sup_{M} |p|, \end{align*} where $M \subset \mathbb{S}^{n-1}$ ($n \ge 2$) is a fractal set of positive $(n-2+\delta)$-Hausdorff content for arbitrary $\delta \in (0,1)$, and $p$ is a spherical polynomial of degree at most $N\in \mathbb{Z}^+$. Second, building upon this fractal framework, we establish sharp observability inequalities for the heat equation on the sphere, again valid for all $\delta\in (0, 1)$, which improve the result of Burq and Moyano [J. Eur. Math. Soc. (JEMS), 25 (4) (2023)] in the spherical setting. Furthermore, as an additional application, we prove a lower-dimensional observability inequality for the heat equation with super-quadratic potentials $V(x) = |x|^{2m}$ ($m \in \mathbb{Z}^+, m\ge 2$) on the whole space $\mathbb{R}^n$.

math.AP↗

arXiv2026

Fractal Tur\'{a}n-Nazarov Inequality and Observability for Schr\"{o}dinger Equations

This paper establishes limitations on observability inequality and unique continuation for Schr\"{o}dinger equations on fractal sets. We prove that, in contrast to the heat equation, such properties can fail in fractal settings. To achieve this, we first extend the classical Tur\'{a}n--Nazarov inequality, which provides lower bounds of trigonometric polynomials of the form $\sum_{k=1}^nc_ke^{2\pi im_kt}$ on sets of positive measure, to the fractal setting. Unlike in the classical case, the constant in the inequality loses uniformity in the degree $n$, and we obtain sharp bounds depending on both $n$ and the frequency difference $m_n-m_1$. These refinements then enable us to construct explicit counterexamples, showing that observability and unique continuation may fail for Schr\"{o}dinger equations when the observation set is fractal.

math.AP↗

arXiv2025

The $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations

This paper investigates the $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations \begin{equation*} H=-\Delta+\sum\limits_{i=1}^N\langle\cdot\,, \varphi_i\rangle \varphi_i \qquad \mbox{on}\,\,\, \R^d. \end{equation*} For dimensions $d\ge 3$, we prove that the wave operators $W_\pm(H,H_0)$ are bounded on $L^p$ for the full range $1\le p\le \infty$. This extends the work of Nier and the third author \cite{NS} by resolving the previously unexplored question of boundedness at the endpoint cases $p=1$ and $p=\infty$. In lower dimensions $d = 1, 2$, we establish the $L^p$-boundedness of the wave operators for the first time. Furthermore, we reveal an intriguing dichotomy in the endpoint case $p = 1$: \begin{itemize} \item If $\int_{\mathbb{R}^d} \varphi_i(x) \, \d x = 0$ holds for every $1\le i\le N$, then the wave operators are bounded on $L^p(\mathbb{R}^d)$ for all $1 \leq p \leq \infty$. \item If there exists at least one $i$ ($1\le i\le N$) such that $\int_{\mathbb{R}^d}\varphi_i(x)\d x\ne0$, then the wave operators remain bounded for $1 < p < \infty$ and satisfy weak type $(1,1)$ estimates, but fail to be bounded on $L^1(\mathbb{R}^d)$. \end{itemize}

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arXiv2025

Dynamical versions of Morgan's Uncertainty Principle and Electromagnetic Schr\"{o}dinger Evolutions

This paper investigates the unique continuation properties of solutions of the electromagnetic Schr\"{o}dinger equation $$ i\partial_{t}u(x,t)+(\nabla-i A)^{2}u(x,t)=V(x,t)u(x,t)\,\,\,\, \mbox{in} \,\,\,\mathbb{R}^{n}\times [0,1], $$ where $A$ represents a time-independent magnetic vector potential and $V$ is a bounded, complex valued time-dependent potential. Given $1 0$ and there exists $N_{p}>0$ such that \begin{equation*} \alpha\beta>N_p, \end{equation*} then $u\equiv 0$. These results can be interpreted as dynamical versions of the uncertainty principle of Morgan's type. Furthermore, as an application, our results extend to a large class of semi-linear Schr\"{o}dinger equations.

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arXiv2025

Pointwise estimates for the fundamental solutions of higher order schr\"{o}dinger equations with finite rank perturbations

This paper is dedicated to studying pointwise estimates of the fundamental solution for the higher order Schr\"{o}dinger equation: % we investigate the fundamental solution of the higher order Schr\"{o}dinger equation $$i{\partial}_{t}u(x,t)=Hu(x,t),\ \ \ t\in \mathbb{R},\ x\in {\mathbb{R}}^{n},$$ where the Hamiltonian $H$ is defined as $$H={(-\Delta)}^{m}+\displaystyle\sum_{j=1}^{N} \langle\cdotp ,{\varphi }_{j} \rangle{\varphi }_{j},$$ with each $\varphi_j$ ($1\le j\le N$) satisfying certain smoothness and decay conditions. %Let ${P}_{ac}(H)$ denote the projection onto the absolutely continuous space of $H$. We show that for any positive integer $m>1$ and spatial dimension $n\ge 1$, %under a spectral assumption, the operator is sharp in the sense that it ${e}^{-i tH}P_{ac}(H)$ has an integral kernel $K(t,x,y)$ satisfying the following pointwise estimate: $$\left |K(t,x,y)\right |\lesssim |t|^{-\frac{n}{2m}}(1+|t|^{-\frac{1}{2m}}\left | x-y\right |)^{-\frac{n(m-1)}{2m-1}} ,\ \ t\ne 0,\ x,y\in {\mathbb{R}}^{n}.$$ This estimate is consistent with the upper bounds in the free case. As an application, we derive $L^p-L^q$ decay estimates for the propagator ${e}^{-\i tH}P_{ac}(H)$, where the pairs $(1/p, 1/q)$ lie within a quadrilateral region in the plane.

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arXiv2025

Quantitative observability for the Schr\"{o}dinger equation with an anharmonic oscillator

This paper studies the observability inequalities for the Schr\"{o}dinger equation associated with an anharmonic oscillator $H=-\frac{\d^2}{\d x^2}+|x|$. We build up the observability inequality over an arbitrarily short time interval $(0,T)$, with an explicit expression for the observation constant $C_{obs}$ in terms of $T$, for some observable set that has a different geometric structure compared to those discussed in \cite{HWW}. We obtain the sufficient conditions and the necessary conditions for observable sets, respectively. We also present counterexamples to demonstrate that half-lines are not observable sets, highlighting a major difference in the geometric properties of observable sets compared to those of Schr\"{o}dinger operators $H=-\frac{\d^2}{\d x^2}+|x|^{2m}$ with $m\ge 1$. Our approach is based on the following ingredients: First, the use of an Ingham-type spectral inequality constructed in this paper; second, the adaptation of a quantitative unique compactness argument, inspired by the work of Bourgain-Burq-Zworski \cite{Bour13}; third, the application of the Szeg\"{o}'s limit theorem from the theory of Toeplitz matrices, which provides a new mathematical tool for proving counterexamples of observability inequalities.

math.AP↗

arXiv2024

Observability inequality, log-type Hausdorff content and heat equations

This paper studies observability inequalities for heat equations on both bounded domains and the whole space $\mathbb{R}^d$. The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content. Notably, the aforementioned inequality holds not only for all sets with Hausdorff dimension $s$ for any $s\in (d-1,d]$, but also for certain sets of Hausdorff dimension $d-1$. On the whole space $\mathbb{R}^d$, we establish the observability inequality for observation sets that are thick at the scale of the log-type Hausdorff content. Furthermore, we prove that for the 1-dimensional heat equation on an interval, the Hausdorff content we have chosen is an optimal scale for the observability inequality. To obtain these observability inequalities, we use the adapted Lebeau-Robiano strategy from \cite{Duyckaerts2012resolvent}. For this purpose, we prove the following results at scale of the log-type Hausdorff content, the former being derived from the latter: We establish a spectral inequality/a Logvinenko-Sereda uncertainty principle; we set up a quantitative propagation of smallness of analytic functions; we build up a Remez' inequality; and more fundamentally, we provide an upper bound for the log-type Hausdorff content of a set where a monic polynomial is small, based on an estimate in Lubinsky \cite{Lubinsky1997small}, which is ultimately traced back to the classical Cartan Lemma. In addition, we set up a capacity-based slicing lemma (related to the log-type gauge functions) and establish a quantitative relationship between Hausdorff contents and capacities. These tools are crucial in the studies of the aforementioned propagation of smallness in high-dimensional situations.

math.AP↗

arXiv2024

Pointwise estimates for the fundamental solutions of higher order Schr\"{o}dinger equations in odd dimensions II: high dimensional case

In this paper, for any odd $n$ and any integer $m\geq1$ with $n>4m$, we study the fundamental solution of the higher order Schr\"{o}dinger equation \begin{equation*} \mathrm{i}\partial_tu(x,t)=((-\Delta)^m+V(x))u(x,t),\quad t\in \mathbb{R},\,\,x\in \mathbb{R}^n, \end{equation*} where $V$ is a real-valued $C^{\frac{n+1}{2}-2m}$ potential with certain decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum space of $H=(-\Delta)^m+V$, and assume that $H$ has no positive embedded eigenvalue. Our main result says that $e^{-\mathrm{i}tH}P_{ac}(H)$ has integral kernel $K(t,x,y)$ satisfying \begin{equation*} |K(t, x,y)|\le C(1+|t|)^{-(\frac{n}{2m}-\sigma)}(1+|t|^{-\frac{n}{2 m}})\left(1+|t|^{-\frac{1}{2 m}}|x-y|\right)^{-\frac{n(m-1)}{2 m-1}},\quad t\neq0,\,x,y\in\mathbb{R}^n, \end{equation*} where $\sigma=2$ if $0$ is an eigenvalue of $H$, and $\sigma=0$ otherwise. A similar result for smoothing operators $H^\frac{\alpha}{2m}e^{-\mathrm{i}tH}P_{ac}(H)$ is also given. The regularity condition $V\in C^{\frac{n+1}{2}-2m}$ is optimal in the second order case, and it also seems optimal when $m>1$.

math.AP↗

arXiv2024

Pointwise estimates for the fundamental solutions of higher order Schr\"{o}dinger equations in low odd dimensions

In this paper, we study the fundamental solution of the higher order Schr\"odinger equation \begin{equation*} \mathrm{i}\partial_t u(x,t) = \big((-\Delta)^m + V(x)\big)u(x,t), \quad t \in \mathbb{R}, \ x \in \mathbb{R}^n, \end{equation*} for any odd dimension $n$ and integer $m \geq 1$ satisfying $n < 4m$, where $V$ is a real-valued bounded potential with suitable decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectral subspace of $H = (-\Delta)^m + V$, and assume $H$ has no positive embedded eigenvalues. Our main result says that the evolution operator $e^{-\mathrm{i}tH}P_{ac}(H)$ has an integral kernel $K(t,x,y)$ satisfying the pointwise estimate \begin{equation*} |K(t,x,y)| \leq C (1 + |t|)^{-h} (1 + |t|^{-\frac{n}{2m}}) \left(1 + |t|^{-\frac{1}{2m}}|x - y|\right)^{-\frac{n(m-1)}{2m-1}}, \quad t \neq 0, \ x,y \in \mathbb{R}^n, \end{equation*} where the exponent $h$ depends on $m$, $n$, and the zero energy resonance structure of $H$. We also prove analogous estimates for smoothing operators of the form $H^{\frac{\alpha}{2m}}e^{-\mathrm{i}tH}P_{ac}(H)$. The key innovation of this paper is a unified approach to deriving asymptotic expansions of the perturbed resolvents around zero, which comprehensively addresses all possible resonance types.

math.AP↗

arXiv2023

Heisenberg Uniqueness Pairs and the wave equation

Given a curve $Γ$ and a set $Λ$ in the plane, the concept of the Heisenberg uniqueness pair $(Γ, Λ)$ was first introduced by Hedenmalm and Motes-Rodr\'ıgez (Ann. of Math. 173(2),1507-1527, 2011, \cite{HM}) as a variant of the uncertainty principle for the Fourier transform. The main results of Hedenmalm and Motes-Rodr\'ıgez concern the hyperbola $Γ_ε=\{(x_1, x_2)\in \mathbb{R}^2,\, x_1x_2=ε\}$ ($0\neε\in \mathbb{R}$) and lattice-crosses $Λ_{αβ}=(α\mathbb{Z}\times \{0\})\cup(\{0\}\times β\mathbb{Z})$ ($α, β>0$), where it's proved that $(Γ_ε, Λ_{αβ})$ is a Heisenberg uniqueness pair if and only if $αβ\leq 1/|ε|$. In this paper, we aim to study the endpoint case (i.e., $ε=0$ in $Γ_ε$) and investigate the following problem: what's the minimal amount of information required on $Λ$ (the zero set) to form a Heisenberg uniqueness pair? When $Λ$ is contained in the union of two curves in the plane, we give characterizations in terms of some dynamical system conditions. The situation is quite different in higher dimensions and we obtain characterizations in the case that $Λ$ is the union of two hyperplanes.

math.CA↗

arXiv2022

Dispersive estimates for the Schrödinger equation with finite rank perturbations

In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$ H=-Δ+\sum_{j=1}^N\langle\cdot\,, φ_j\rangle φ_j\quad\,\,\,\text{in}\,\,\,\mathbb{R}^d,\,\, d\ge 1, $$ where each $φ_j$ satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant $C=C(N, d, φ_1,\ldots, φ_N)>0$ such that $$ \|e^{-itH}\|_{L^1-L^{\infty}}\leq C t^{-\frac{d}{2}}, \,\,\,\text{for}\,\,\, t>0. $$ As far as we are aware, this seems to provide the first study of $L^1-L^{\infty}$ estimates for finite rank perturbations of the Laplacian in any dimension. We first deal with rank one perturbations ($N=1$). Then we turn to the general case. The new idea in our approach is to establish the Aronszajn-Krein type formula for finite rank perturbations. This allows us to reduce the analysis to the rank one case and solve the problem in a unified manner. Moreover, we show that in some specific situations, the constant $C(N, d, φ_1,\ldots, φ_N)$ grows polynomially in $N$. Finally, as an application, we are able to extend the results to $N=\infty$ and deal with some trace class perturbations.

math.AP↗

arXiv2021

Unique continuation properties for one dimensional higher order Schrödinger equations

We study two types of unique continuation properties for the higher order Schrödinger equation with potential $$ i\partial_tu=(-Δ_x)^mu+V(t,x)u,\quad(t,x)\in\mathbb{R}^{1+n},\,2\leq m\in\mathbb{N}_+. $$ The first one says if $u$ has certain exponential decay at two times, then $u\equiv0$, and this result is sharp by constructing critical non-trivial solutions. The second one says if $u\equiv0$ in an arbitrary half-space of $\mathbb{R}^{1+n}$, then $u\equiv0$ identically. The uniqueness theorems are given when $n=1$, but we also prove partial results when $n\in\mathbb{N}_+$ for their own interests. Possibility or obstacles to proving these unique continuation properties in higher spatial dimensions are also discussed.

math.AP↗

arXiv2020

Characterizations of stabilizable sets for some parabolic equations in $\mathbb{R}^n$

We consider the parabolic type equation in $\mathbb{R}^n$: \begin{align}\label{equ-0} (\partial_t+H)y(t,x)=0,\,\,\, (t,x)\in (0,\infty)\times\mathbb{R}^n;\;\; \quad y(0,x)\in L^2(\mathbb{R}^n), \end{align} where $H$ can be one of the following operators: (i) a shifted fractional Laplacian; (ii) a shifted Hermite operator; (iii) the Schrödinger operator with some general potentials. We call a subset $E\subset \mathbb{R}^n$ as a stabilizable set for the above equation, if there is a linear bounded operator $K$ on $L^2(\mathbb{R}^n)$ so that the semigroup $\{e^{-t(H-χ_EK)}\}_{t\geq 0}$ is exponentially stable. (Here, $χ_E$ denotes the characteristic function of $E$, which is treated as a linear operator on $L^2(\mathbb{R}^n)$.) This paper presents different geometric characterizations of the stabilizable sets for the above equation with different $H$. In particular, when $H$ is a shifted fractional Laplacian, $E\subset \mathbb{R}^n$ is a stabilizable set if and only if $E\subset \mathbb{R}^n$ is a thick set, while when $H$ is a shifted Hermite operator, $E\subset \mathbb{R}^n$ is a stabilizable set for if and only if $E\subset \mathbb{R}^n$ is a set of positive measure. Our results, together with the results on the observable sets for the above equation obtained in \cite{AB,Ko,Li,M09}, reveal such phenomena: for some $H$, the class of stabilizable sets contains strictly the class of observable sets, while for some other $H$, the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for the above equation where $H$ is the Schrödinger operator with some general potentials.

math.AP↗

arXiv2020

Observable sets, potentials and Schrödinger equations

We characterize observable sets for 1-dim Schrödinger equations in $\mathbb{R}$: $i \partial_t u = (-\partial_x^2+x^{2m})u$ (with $m\in \mathbb{N}:=\{0,1,\dots\}$). More precisely, we obtain what follows: First, when $m=0$, $E\subset\mathbb{R}$ is an observable set at some time if and only if it is thick, namely, there is $γ>0$ and $L>0$ so that $$ \left|E \bigcap [x, x+ L]\right|\geq γL\;\;\mbox{for each}\;\;x\in \mathbb{R}; $$ Second, when $m=1$ ($m\geq 2$ resp.), $E$ is an observable set at some time (at any time resp. ) if and only if it is weakly thick, namely $$ \varliminf_{x \rightarrow +\infty} \frac{|E\bigcap [-x, x]|}{x} >0. $$ From these, we see how potentials $x^{2m}$ affect the observability (including the geometric structures of observable sets and the minimal observable time). Besides, we obtain several supplemental theorems for the above results, in particular, we find that a half line is an observable set at time $T>0$ for the above equation with $m=1$ if and only if $T>\fracπ{2}$.

math.OC↗

arXiv2019

Uncertainty principle, minimal escape velocities and observability inequalities for schrödinger equations

We develop a new abstract derivation of the observability inequalities at two points in time for Schrödinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\mathbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular, minimal velocity estimates introduced by Sigal and Soffer. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schrödinger equation.

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↗

arXiv2017

Remarks on $L^p$-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems

This paper comprises two parts. We first investigate a $L^p$ type of limiting absorption principle for Schrödinger operators $H=-Δ+V$, i.e., In $\mathbb{R}^n$ ($n\ge 3$) we prove the $ε-$uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ estimates of the resolvent $(H-λ\pm iε)^{-1}$ for all $λ>0$ when the potential $V$ belongs to some integrable spaces and a spectral condition of $H$ at zero is assumed. As an application, we establish a sharp spectral multiplier theorem and $L^p$ bound of Bochner-Riesz means associated with Schrödinger operators $H$. Next, we consider the fractional Schrödinger operator $H=(-Δ)^α+V$ ($0<2α<n$) and prove a uniform Hardy-Littlewood-Sobolev inequality for $(-Δ)^α$, which generalizes the corresponding result of Kenig-Ruiz-Sogge \cite{KRS}.

math.AP↗

arXiv2017

Inhomogeneous Oscillatory Integrals and Global Smoothing Effects for Dispersive Equations

We study oscillatory integrals of the type ${\mathcal F}^{-1}(e^{ita(\cdot)}ψ(\cdot))$ where $a$ is a general function satisfying some elliptic type and non-degenerate conditions at both the origin and infinity, and $ψ$ belongs to some symbol class. Point-wise estimates in space-time are gained with partial sharpness. As applications, global smoothing effects of $L^p-L^q$ as well as Strichartz type for dispersive equations are studied. An application to fractional Schrödinger equations is also given.

math.AP↗