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Zijun Wan

Publications and source records attributed to Zijun Wan.

9 recordsLinked to original sources
arXiv2026

Time-Decay Estimates for Two-Dimensional Fourth-Order Schr\"odinger Operators with Threshold Singularities

We establish time-decay estimates for the two-dimensional fourth-order Schr\"odinger operator $H=\Delta^2+V$ with a real-valued decaying potential $V$, covering all possible zero-energy threshold obstructions. When zero is a regular point or a first-kind resonance, we prove \[ \left\| H^{\frac{\alpha}{4}}e^{-itH}P_{\mathrm{ac}}(H) \right\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{2+\alpha}{4}}, \qquad -2<\alpha\leq2, \] which matches with the free sharp decay rate throughout the full range of $\alpha$. For a second-kind resonance, the decay rate is $|t|^{-(2+\alpha)/4}(\log(2+|t|))^2$ for every $-2<\alpha\leq2$, with only a logarithmic loss. For the stronger threshold singularities, we show that the large-time behavior is governed by the presence of a \(d\)-wave resonance. If zero is a third-kind resonance, or an eigenvalue accompanied by a \(d\)-wave resonance, we obtain the sharp decay $(\log|t|)^{-1}$ for $\alpha=0$ and $|t|^{-\alpha/4}(\log|t|)^{-2}$ for $0<\alpha\leq2$. If zero is an eigenvalue without a $d$-wave resonance, the second-kind estimate is recovered for $-2<\alpha\leq2$. In addition, in the regular and first-kind resonance cases, we obtainthe logarithmically improved weighted estimate for every $2<\alpha\leq2$ and $s>0$: \[ \left\| \omega^{-s} H^{\frac{\alpha}{4}}e^{-itH}P_{\mathrm{ac}}(H)\omega^{-s} \right\|_{L^1\to L^\infty} \lesssim \frac{1} {|t|^{\frac{2+\alpha}{4}}(\log|t|)^s}, \qquad |t|\geq2, \] where $\omega(x)=\log(2+|x|)$. By contrast, zero is a second-kind resonance for the free operator $\Delta^2$, and the free evolution admits no such logarithmic gain. Thus, in the regular and first-kind cases, the potential changes the zero-energy spectral structure of the free operator, and this change is accompanied by improved weighted decay.

math.AP↗
arXiv2026

Decay estimates for the two-dimensional Beam equation with potentials

This paper establishes time decay estimates for the following two-dimensional beam (plate) equation with a decaying real-valued potential $V$: \begin{equation*} \partial_t^2 u + (\Delta^2 + V) u = 0, \qquad u(0,x)=f(x),\quad \partial_t u(0,x)=g(x). \end{equation*} When zero is a regular point or a first-kind resonance of $H=\Delta^2+V$, we first prove sharp $L^1\to L^\infty$ estimates for the solution operators: \begin{align*} \left\|\cos(t\sqrt{H})P_{\mathrm{ac}}(H)\right\|_{L^1\to L^\infty} + \left\|\frac{\sin(t\sqrt{H})}{t\sqrt{H}}P_{\mathrm{ac}}(H)\right\|_{L^1\to L^\infty} \lesssim \frac{1}{|t|}, \end{align*} and obtain an enhanced decay $(|t|\log|t|)^{-1}$ in logarithmically weighted spaces $L^1_\omega\to L^\infty_{-\omega}$ with $\omega(x)=\log(2+|x|)$. For second-kind resonances of $H$ (the bi-Laplacian $\Delta^2$ belongs to this class), a non-zero trace moment $\langle |x|^2V,\phi\rangle\neq0$ for some second-kind resonance function $\phi$ induces severe threshold singularities, worsening the $L^1\to L^\infty$ estimate to $|t|^{-1}(\log|t|)^2$. Finally, for third-kind resonances or a zero eigenvalue, we prove that the presence of $d$-wave resonance leads to the worst $L^1\to L^\infty$ decay rate $\sim(\log|t|)^{-1}$. Several improved estimates are also obtained without a $d$-wave resonance. In particular, in the pure eigenvalue case (i.e., neither $d$-wave nor $p$-wave resonance), both propagators recover the optimal unweighted $L^1\to L^\infty$ estimate $|t|^{-1}.$

math.AP↗
arXiv2025

The $L^p$-boundedness of wave operators for nonhomogeneous fourth-order Schr\"odinger operators in high dimensions

This paper investigates the $L^p$-boundedness of wave operators associated with the nonhomogeneous fourth-order Sch\"odinger operator $H = \Delta^2 - \Delta + V(x)$ on $\mathbb{R}^n$. Assuming the real-valued potential $ V $ exhibits sufficient decay and regularity, we prove that for all dimensions $ n \geq 5 $, the wave operators $ W_{\pm}(H, H_0)$ are bounded on $L^{p}(\mathbb{R}^{n}) $ for all $ 1 \leq p \leq \infty $, provided that zero is a regular threshold of $H $. As applications, we derive the sharp $L^p$-$L^{p'}$ dispersive estimates for Schr\"odinger group $e^{-itH}$, as well as for the solutions operators $\cos(t \sqrt{H})$ and $\frac{\sin (t \sqrt{H})}{ \sqrt{H}}$ associated with the following beam equations with potentials: $$ \partial_t^2 u + \left(\Delta^2 -\Delta+ V(x) \right) u = 0, \ \ u(0, x) = f(x), \quad \partial_t u(0, x) = g(x),\ \ (t, x) \in \mathbb{R} \times \mathbb{R}^n,\ n\geq5, $$ where $p'$ denotes the H\"older conjugate of $p$, with $1 \leq p \leq 2$. Moreover, we remark that the same results hold for the operator $ \epsilon \Delta^2 - \Delta + V$ with a parameter $\epsilon>0,$ providing greater flexibility for the analysis of related equations.

math.AP↗
arXiv2024

Decay estimates for beam equations with potentials on the line

This paper is devoted to the time decay estimates for the following beam equation with a potential on the line: $$ \partial_t^2 u + \left( \Delta^2 + m^2 + V(x) \right) u = 0, \ \ u(0, x) = f(x),\quad \partial_t u(0, x) = g(x), $$ where $V$ is a real-valued decaying potential on $\mathbb{R}$, and $m \in \mathbb{R}$. Let $H = \Delta^2 + V$ and $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum of $H$. Then for $m = 0$, we establish the following decay estimates of the solution operators: $$ \left\|\cos (t \sqrt{H}) P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} + \left\|\frac{\sin (t \sqrt{H})}{t \sqrt{H}} P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} \lesssim |t|^{-\frac{1}{2}}. $$ But for $m \neq 0$, the solutions have different time decay estimates from the case where $m=0$. Specifically, the $L^1$-$L^\infty$ estimates of $\cos (t \sqrt{H + m^2})$ and $\frac{\sin (t \sqrt{H + m^2})}{\sqrt{H + m^2}}$ are bounded by $O(|t|^{-\frac{1}{4}})$ in the low-energy part and $O(|t|^{-\frac{1}{2}})$ in the high-energy part. It is noteworthy that all these results remain consistent with the free cases (i.e., $V = 0$) whatever zero is a regular point or a resonance of $H$. As consequences, we establish the corresponding Strichartz estimates, which are fundamental to study nonlinear problems of beam equations.

math.AP↗
arXiv2024

Breaking the Midas Spell:Understanding Progressive Novice-AI Collaboration in Spatial Design

In spatial design, Artificial Intelligence (AI) tools often generate the entire spatial design outcome in a single automated step, rather than engaging users in a deepening and iterative process. This significantly reduces users' involvement, learning, and creative capabilities, leading to a superficial understanding of spatial design. We conducted a Wizard-of-Oz study, where Novices and AI (acted by experimenters) worked together to finish spatial design tasks using various AI models. We identified typical function and workflow patterns adopted by the participants, leading to the understanding of the opportunities and challenges in the human-AI co-creation process. Based on insights gathered from this research, we proposed some design implications of the novice-AI collaboration system that aims to democratize spatial design through a progressive, iterative co-creation process.

cs.HC↗
arXiv2023

$L^p$-boundedness of wave operators for fourth order Schr\"odinger operators with zero resonances on $\mathbb{R}^3$

Let $H = \Delta^2 + V$ be the fourth-order Schr\"odinger operator on $\mathbb{R}^3$ with a real-valued fast-decaying potential $V$. If zero is neither a resonance nor an eigenvalue of $H$, then it was recently shown that the wave operators $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < \infty$ and unbounded at the endpoints $p=1$ and $p=\infty$. This paper is to further establish the $L^p$-boundedness of $W_\pm(H, \Delta^2)$ that exhibit all types of singularities at the zero energy threshold. We first prove that $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < \infty$ in the first kind resonance case, and then proceed to establish for the second kind resonance case that they are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < 3$, but not if $3 \le p \le \infty$. In the third kind resonance case, we also show that $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1<p<3$ and generically unbounded on $L^p(\R^3)$ for any $3\le p\le\infty$. Moreover, it is also shown that $W_\pm(H, \Delta^2)$ are bounded on $L^p(\R^3)$ for all $3\le p<\infty$ if in addition $H$ has the zero eigenvalue, but no $p$-wave zero resonances and all zero eigenfunctions are orthogonal to $x_ix_jx_kV$ in $L^2(\R^3)$ for all $i,j,k=1,2,3$ with $x=(x_1,x_2,x_3)\in \R^3$. These results describe precisely the validity of the $L^p$-boundedness of $W_\pm(H, \Delta^2)$ in $\mathbb{R}^3$ for all types of singularities at the zero energy threshold with some exceptions for the endpoint cases $p=1,\infty$. As an application, $L^p$-$L^q$ decay estimates are also derived for the fourth-order Schr\"odinger equations and Beam equations with zero resonance singularities.

math.AP↗
arXiv2023

Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schr\"odinger operators in dimension three

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,\Delta^2)$ associated with fourth-order Schr\"odinger operators $H=\Delta^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-\mu}$ for some $\mu>0$. A recent work by Goldberg and Green \cite{GoGr21} has demonstrated that wave operators $W_\pm(H,\Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 9$, and zero is a regular point of $H$. In this paper, we aim to further establish endpoint estimates for $W_\pm(H,\Delta^2)$ in two significant ways. First, we provide counterexamples that illustrate the unboundedness of $W_\pm(H,\Delta^2)$ on the endpoint spaces $L^1(\mathbb{R}^3)$ and $L^\infty(\mathbb{R}^3)$, even for non-zero compactly supported potentials $V$. Second, we establish weak (1,1) estimates for the wave operators $W_\pm(H,\Delta^2)$ and their dual operators $W_\pm(H,\Delta^2)^*$ in the case where zero is a regular point and $\mu>11$. These estimates depend critically on the singular integral theory of Calder\'on-Zygmund on a homogeneous space $(X,d\omega)$ with a doubling measure $d\omega$.

math.AP↗
arXiv2023

Emergent Bio-Functional Similarities in a Cortical-Spike-Train-Decoding Spiking Neural Network Facilitate Predictions of Neural Computation

Despite its better bio-plausibility, goal-driven spiking neural network (SNN) has not achieved applicable performance for classifying biological spike trains, and showed little bio-functional similarities compared to traditional artificial neural networks. In this study, we proposed the motorSRNN, a recurrent SNN topologically inspired by the neural motor circuit of primates. By employing the motorSRNN in decoding spike trains from the primary motor cortex of monkeys, we achieved a good balance between classification accuracy and energy consumption. The motorSRNN communicated with the input by capturing and cultivating more cosine-tuning, an essential property of neurons in the motor cortex, and maintained its stability during training. Such training-induced cultivation and persistency of cosine-tuning was also observed in our monkeys. Moreover, the motorSRNN produced additional bio-functional similarities at the single-neuron, population, and circuit levels, demonstrating biological authenticity. Thereby, ablation studies on motorSRNN have suggested long-term stable feedback synapses contribute to the training-induced cultivation in the motor cortex. Besides these novel findings and predictions, we offer a new framework for building authentic models of neural computation.

q-bio.NC↗
arXiv2022

$L^p$-boundedness of wave operators for bi-Schr\"odinger operators on the line

This paper is devoted to establishing several types of $L^p$-boundedness of wave operators $W_\pm=W_\pm(H, \Delta^2)$ associated with the bi-Schr\"odinger operators $H=\Delta^{2}+V(x)$ on the line $\mathbb{R}$. Given suitable decay potentials $V$, we firstly prove that the wave and dual wave operators are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$: $$ \|W_\pm f\|_{L^p(\mathbb{R})}+\|W_\pm^* f\|_{L^p(\mathbb{R})}\lesssim \|f\|_{L^p(\mathbb{R})},$$ which are further extended to the $L^p$-boundedness on the weighted spaces $L^p(\mathbb{R},w)$ with general even $A_p$-weights $w$ and to the boundedness on the Sobolev spaces $W^{s,p}(\mathbb{R})$. For the limiting case, we prove that $W_\pm$ are bounded from $L^1(\R)$ to $L^{1,\infty}(\R)$ as well as bounded from the Hardy space $\H^1(\R)$ to $L^1(\R)$. These results especially hold whatever the zero energy is a regular point or a resonance of $H$. We also obtain that $W_\pm$ are bounded from $L^\infty(\R)$ to $\BMO(\R)$ if zero is a regular point or a first kind resonance of $H$. Next, we show that $W_\pm$ are neither bounded on $L^1(\mathbb{R})$ nor on $L^\infty(\mathbb{R})$ even if zero is a regular point of $H$. Moreover, if zero is a second kind resonance of $H$, then $W_\pm$ are shown to be even not bounded from $L^\infty(\R)$ to $\BMO(\R)$ in general. In particular, we remark that our results give a complete picture of the validity of $L^p$-boundedness of the wave operators for all $1\le p\le \infty$ in the regular case. Finally, as applications, we deduce the $L^p$-$L^q$ decay estimates for the propagator $e^{-itH}P_{\mathrm{ac}}(H)$ with pairs $(1/p,1/q)$ belonging to a certain region of $\mathbb{R}^2$, as well as establish the H\"ormander-type $L^p$-boundedness theorem for the spectral multiplier $f(H)$.

math.AP↗