arXiv subjects

Sisi Huang

Publications and source records attributed to Sisi Huang.

5 recordsLinked to original sources

arXiv2026

Counterexamples to the $L^1$ and $L^{\infty}$ boundedness of the one-dimensional wave operators

It is well established that the wave operators $W_{\pm}(H,-\Delta)$ for the one-dimensional Schr\"{o}dinger operator $H=-\Delta+V(x)$ are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$ in both generic and exceptional cases. They are also bounded on $L^1(\mathbb{R})$ and $L^{\infty}(\mathbb{R})$ in the exceptional case with $\lim\limits_{x\rightarrow-\infty}f_+(0,x)=1$. For the remaining endpoint cases, it has long been expected that they are generally unbounded at the endpoints $p=1,\infty$ due to the presence of the Hilbert transform in the low energy part, yet a rigorous proof has been missing. In this paper, we show that even for a bounded and compactly supported non-zero potential $V$, the wave operators $W_{\pm}(H,-\Delta)$ are unbounded on $L^1(\mathbb{R})$ and $L^{\infty}(\mathbb{R})$ in the generic case, as well as in the exceptional case with the condition $\lim\limits_{x\rightarrow-\infty}f_+(0,x)\neq1$. Moreover, in the latter case, they are even unbounded from $L^{\infty}(\mathbb{R})$ to ${\rm BMO}(\mathbb{R})$ (Bounded Mean Oscillation space). Hence together with those known results, our counterexamples complete the picture of the $L^{p}$ boundedness of one-dimensional wave operators.

math-ph↗

arXiv2025

The $\ell^p$-boundedness of wave operators for the fourth order Schr\"{o}dinger operators on the lattice $\mathbb{Z}$

This paper investigates the $\ell^p$ boundedness of wave operators $W_\pm(H,\Delta^2)$ associated with discrete fourth-order Schr\"odinger operators $H = \Delta^2 + V$ on the lattice $\mathbb{Z}$, where $$(\Delta\phi)(n)=\phi(n+1)+\phi(n-1)-2\phi(n),\quad n\in\mathbb{Z},$$ and $V(n)$ is a real-valued potential on $\mathbb{Z}$. Under suitable decay assumptions on $V$ (depending on the types of zero resonance of $H$), we show that the wave operators $W_{\pm}(H, \Delta^2)$ are bounded on $\ell^p(\mathbb{Z})$ for all $1 < p < \infty$: $$ \|W_{\pm}(H, \Delta^2) f\|_{\ell^p(\mathbb{Z})} \lesssim \|f\|_{\ell^p(\mathbb{Z})}. $$ In particular, if both thresholds $0$ and $16$ are regular points of $H$, we prove that $W_{\pm}(H, \Delta^2)$ are neither bounded on the endpoint space $\ell^1(\mathbb{Z})$ nor on $\ell^\infty(\mathbb{Z})$. We remark that the proof of these bounds relies fundamentally on the asymptotic expansions of the resolvent of $H$ near the thresholds $0$ and $16$, and on the theory of {\it discrete singular integrals} on the lattice. As applications, we derive the following sharp $\ell^p-\ell^{p'}$ decay estimates for solutions to the discrete beam equation with a parameter $a\in \mathbb{R}$ on the lattice $\mathbb{Z}$: $$ \|{\rm cos}(t\sqrt {H+a^2})P_{ac}(H)\|_{\ell^p\rightarrow\ell^{p'}}+\left\|\frac{{\rm sin}(t\sqrt {H+a^2})}{t\sqrt {H+a^2}}P_{ac}(H)\right\|_{\ell^p\rightarrow\ell^{p'}}\lesssim|t|^{-\frac{1}{3}(\frac{1}{p}-\frac{1}{p'})},\quad t\neq0, $$ where $1<p\le 2$, ${p'}$ is the conjugated index of $p$ and $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$.

math.AP↗

arXiv2025

Decay estimates for discrete bi-Laplace operators with potentials on the lattice $\mathbb{Z}$

It is known that the discrete Laplace operator $\Delta$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\big\|e^{it\Delta}\big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0,$$ which is slower than the usual $ O(|t|^{-\frac{1}{2}})$ decay in the continuous case on $\mathbb{R}$. However, this paper shows that the discrete bi-Laplacian $\Delta^2$ on $\mathbb{Z}$ actually exhibits the same sharp decay estimate $|t|^{-\frac{1}{4}}$ as its continuous counterpart. In view of the free decay estimate, we further investigate the discrete bi-Schr\"{o}dinger operators of the form $H=\Delta^2+V$ on the lattice space $\ell^2(\mathbb{Z})$, where $V$ is a class of real-valued decaying potentials on $\mathbb{Z}$. First, we establish the limiting absorption principle for $H$, and then derive the full asymptotic expansions of the resolvent of $H$ near the thresholds $0$ and $16$, including resonance cases. In particular, we provide a complete characterizations of the different resonance types in $\ell^2$-weighted spaces. Based on these results above, we establish the following sharp $\ell^1-\ell^{\infty}$ decay estimates for all different resonances types of $H$ under suitable decay conditions on $V$: $$\big\|e^{-itH}P_{ac}(H)\big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{4}},\quad t\neq0,$$ where $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$. Additionally, the decay estimates for the evolution flow of discrete beam equation are also derived: $$\|{\cos}(t\sqrt H)P_{ac}(H)\|_{\ell^1\rightarrow\ell^{\infty}}+\Big\|\frac{{\sin}(t\sqrt H)}{t\sqrt H}P_{ac}(H)\Big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0.$$

math.AP↗

arXiv2025

Decay estimates for discrete bi-Schr\"{o}dinger operators on the lattice $\mathbb{Z}$

It is known that the discrete Laplace operator $\Delta$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\left\|e^{it\Delta}\right\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0,$$ which is slower than the usual $|t|^{-\frac{1}{2}}$ decay in the continuous case on $\mathbb{R}$. However in this paper, we have showed that the discrete bi-Laplacian $\Delta^2$ on $\mathbb{Z}$ actually exhibits the same sharp decay estimate $|t|^{-\frac{1}{4}}$ as its continuous counterpart. In view of these free decay estimates, this paper further investigates the discrete bi-Schr\"{o}dinger operators of the form $H=\Delta^2+V$ on the lattice space $\ell^2(\mathbb{Z})$, where $V(n)$ is a real valued potential of $\mathbb{Z}$. Under suitable decay conditions on $V$ and assuming that both 0 and 16 are regular spectral points of $H$, we establish the following sharp $\ell^1-\ell^{\infty}$ dispersive estimates: $$\left\|e^{-itH}P_{ac}(H)\right\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{4}},\quad t\neq0,$$ where $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$. Additionally, the following decay estimates for beam equation are also derived: $$\|{\rm cos}(t\sqrt H)P_{ac}(H)\|_{\ell^1\rightarrow\ell^{\infty}}+\left\|\frac{{\rm sin}(t\sqrt H)}{t\sqrt H}P_{ac}(H)\right\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0.$$

math.AP↗

arXiv2020

Controllability of two-point boundary value problem for wave equations in $L^1$ and $L^2$ spaces: One dimensional case

In this paper we discuss the controllability of two-point boundary value problem (TBVP) for one-dimensional wave equation. Some new concepts are introduced: TBVP input control problem, minimum-input solution (MS) and pre-minimum-input solution (PMS). We set the metric in $L^1$ and $L^2$ spaces on a closed set, and control the input to reach its minimum. And we mainly discuss the property of input, the existence and uniqueness of MS and PMS for $L^1$ and $L^2$ metric respectively. The minimum inputs lie on a strip in $L^1$ and PMS for $L^1$ and $L^2$ always exists. Furthermore, to construct PMS, we also introduce an approximation method which meets certain conditions.

math.AP↗