arXiv subjects

Xiaohua Yao

Publications and source records attributed to Xiaohua Yao.

At least 19 recordsLinked to original sources

arXiv2026

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

It is known that the discrete Laplace operator $Δ$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\big\|e^{itΔ}\big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0,$$ 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 $Δ^2$ 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-Schrödinger operators of the form $H=Δ^2+V$ 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↗

arXiv2026

Endpoint Mapping Properties of Wave Operators for Schrödinger Operators in Dimensions $n\ge3$

We study endpoint mapping properties of low-energy wave operators for Schrödinger operators $H=-Δ+V$ on $\mathbb R^n$, $n\ge3$. In dimension four, we prove that a zero-energy resonance prevents $L^1$ boundedness, whether or not zero is also an eigenvalue, under $|V(x)|\lesssim\langle x\rangle^{-β}$ with $β>10$. For a zero-energy eigenvalue without a resonance, we obtain a complete low-energy $L^p$ classification in every dimension $n\ge3$ under $β>n+4$. In particular, $L^\infty$ boundedness is equivalent to the vanishing of the zeroth, first, and harmonic second moments of $Vψ$ for every zero-energy eigenfunction $ψ$. The proof identifies the finite-rank obstruction and shows that the remaining eigenvalue correction cannot cancel its critical asymptotic profiles. The same conclusions hold for the full wave operators when the corresponding high-energy bounds are available.

math.AP↗

arXiv2026

Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schrödinger Operators

We establish sharp endpoint mapping properties for the wave operators $W_\pm(H,-Δ)$ of two-dimensional Schrödinger operators $H=-Δ+V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints $p=1$ and $p=\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1(\mathbb{R}^2)$ and on $L^\infty(\mathbb{R}^2)$, but they satisfy the atural substitute estimates of Calderón--Zygmund type: $$ L^1(\mathbb{R}^2)\longrightarrow L^{1,\infty}(\mathbb{R}^2),\ \ \ \mathcal{H}^1(\mathbb{R}^2)\longrightarrow L^1(\mathbb{R}^2),\ \ \ L^\infty(\mathbb{R}^2)\longrightarrow \mathrm{BMO}(\mathbb{R}^2). $$ When zero is instead a threshold singularity of the first kind---an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces $L^1(\mathbb{R}^2)$ and $L^\infty(\mathbb{R}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\ge3$, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of $H$. A p-wave resonance obstructs both the $L^1$- and the $L^\infty$-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.

math.AP↗

arXiv2026

Time-Decay Estimates for Two-Dimensional Fourth-Order Schrödinger Operators with Threshold Singularities

We establish time-decay estimates for the two-dimensional fourth-order Schrödinger operator $H=Δ^2+V$ 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 \[ \left\| H^{\fracα{4}}e^{-itH}P_{\mathrm{ac}}(H) \right\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{2+α}{4}}, \qquad -2<α\leq2, \] which matches with the free sharp decay rate throughout the full range of $α$. For a second-kind resonance, the decay rate is $|t|^{-(2+α)/4}(\log(2+|t|))^2$ for every $-2<α\leq2$, 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 $α=0$ and $|t|^{-α/4}(\log|t|)^{-2}$ for $0<α\leq2$. If zero is an eigenvalue without a $d$-wave resonance, the second-kind estimate is recovered for $-2<α\leq2$. In addition, in the regular and first-kind resonance cases, we obtainthe logarithmically improved weighted estimate for every $2<α\leq2$ and $s>0$: \[ \left\| ω^{-s} H^{\fracα{4}}e^{-itH}P_{\mathrm{ac}}(H)ω^{-s} \right\|_{L^1\to L^\infty} \lesssim \frac{1} {|t|^{\frac{2+α}{4}}(\log|t|)^s}, \qquad |t|\geq2, \] where $ω(x)=\log(2+|x|)$. By contrast, zero is a second-kind resonance for the free operator $Δ^2$, 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 + (Δ^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 $H=Δ^2+V$, 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 $L^1_ω\to L^\infty_{-ω}$ with $ω(x)=\log(2+|x|)$. For second-kind resonances of $H$ (the bi-Laplacian $Δ^2$ belongs to this class), a non-zero trace moment $\langle |x|^2V,ϕ\rangle\neq0$ for some second-kind resonance function $ϕ$ 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↗

arXiv2026

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

It is well established that the wave operators $W_{\pm}(H,-Δ)$ for the one-dimensional Schrödinger operator $H=-Δ+V(x)$ 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 $W_{\pm}(H,-Δ)$ 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 Schrödinger operators on the lattice $\mathbb{Z}$

This paper investigates the $\ell^p$ boundedness of wave operators $W_\pm(H,Δ^2)$ associated with discrete fourth-order Schrödinger operators $H = Δ^2 + V$ on the lattice $\mathbb{Z}$, where $$(Δϕ)(n)=ϕ(n+1)+ϕ(n-1)-2ϕ(n),\quad n\in\mathbb{Z},$$ 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 $W_{\pm}(H, Δ^2)$ are bounded on $\ell^p(\mathbb{Z})$ for all $1 < p < \infty$: $$ \|W_{\pm}(H, Δ^2) f\|_{\ell^p(\mathbb{Z})} \lesssim \|f\|_{\ell^p(\mathbb{Z})}. $$ In particular, if both thresholds $0$ and $16$ are regular points of $H$, we prove that $W_{\pm}(H, Δ^2)$ 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

The $L^p$-boundedness of wave operators for higher order Schrödinger operator with zero singularities in low odd dimensions

This paper investigates the $L^p$-bounds of wave operators for higher-order Schrödinger operators $H = (-Δ)^m + V$ on $\mathbb{R}^n$, with $m \ge 2$ and real-valued decaying potentials $V$. Our main objective is to establish the sharp $L^p$-boundedness of the wave operators $W_\pm(H; (-Δ)^m)$ in the presence of all types of zero-resonance singularities, for all odd dimensions $1 \le n \le 4m - 1$. Specifically, for odd $n$ with $1 \le n \le 4m - 1$, there exist $m_n$ types of zero resonances for $H$, along with a critical type $k_c$ (both depending on $n$ and $m$). If zero is a regular point of $H$ or a $\mathbf{k}$-th kind resonance with $1 \le \mathbf{k} \le k_c$, the wave operators $W_\pm(H; (-Δ)^m)$ are bounded on $L^p(\mathbb{R}^n)$ for all $1 < p < \infty$. If zero is a $\mathbf{k}$-th kind resonance with $k_c < \mathbf{k} \le m_n$, we show that the range of $p$-boundedness for $W_\pm(H; (-Δ)^m)$ narrows to $1 < p < p_{\mathbf{k}}$, where $$p_{\mathbf{k}} = \frac{n}{n - 2m + \mathbf{k} + k_c - 1}.$$ Additionally, if zero is an eigenvalue of $H$ (i.e., $\mathbf{k} = m_n + 1$), then $W_\pm(H; (-Δ)^m)$ are bounded on $L^p(\mathbb{R}^n)$ for all $1 < p < \frac{2n}{n - 1}$. Furthermore, it is shown that the wave operators $W_\pm(H; (-Δ)^m)$ are unbounded on $L^p(\mathbb{R}^n)$ for all $p_{\mathbf{k}} < p \le \infty$ if $k_c < \mathbf{k} \le m_n$, and for all $\frac{2n}{n - 1} < p \le \infty$ if zero is an eigenvalue of $H$ with a non-zero solution $ϕ$ to $Hϕ= 0$ in $\bigcap_{s < -\frac{1}{2}} L^{2}_{s}(\mathbb{R}^n) \setminus L^2(\mathbb{R}^n)$(referred to as a $p$-wave resonance). The key idea of the proof is to reduce the $L^p$-unboundedness to establishing the optimality of time-decay estimates for $e^{itH}P_{ac}(H)$ in weighted $L^2$ spaces.

math.AP↗

arXiv2025

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

Let $H = Δ^2 + V$ be the fourth-order Schrödinger 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 $W_\pm(H, Δ^2)$ 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 $W_\pm(H, Δ^2)$ that exhibit all types of singularities at the zero energy threshold. We first prove that $W_\pm(H, Δ^2)$ 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 $W_\pm(H, Δ^2)$ 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 $W_\pm(H, Δ^2)$ 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 $W_\pm(H, Δ^2)$ 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 Schrödinger equations and Beam equations with zero resonance singularities.

math.AP↗

arXiv2025

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: $$ \partial_t^2 u + \left( Δ^2 + m^2 + V(x) \right) u = 0, \ \ u(0, x) = f(x),\quad \partial_t u(0, x) = g(x), $$ where $V$ is a real-valued decaying potential on $\mathbb{R}$, and $m \in \mathbb{R}$. Let $H = Δ^2 + V$ 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↗

arXiv2025

The $L^p$-boundedness of wave operators for nonhomogeneous fourth-order Schrödinger operators in high dimensions

This paper investigates the $L^p$-boundedness of wave operators associated with the nonhomogeneous fourth-order Schödinger operator $H = Δ^2 - Δ+ V(x)$ 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 Schrödinger 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: $$ \partial_t^2 u + \left(Δ^2 -Δ+ V(x) \right) u = 0, \ \ u(0, x) = f(x), \quad \partial_t u(0, x) = g(x),\ \ (t, x) \in \mathbb{R} \times \mathbb{R}^n,\ n\geq5, $$ where $p'$ denotes the Hölder conjugate of $p$, with $1 \leq p \leq 2$. Moreover, we remark that the same results hold for the operator $ εΔ^2 - Δ+ V$ with a parameter $ε>0,$ providing greater flexibility for the analysis of related equations.

math.AP↗

arXiv2025

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

It is known that the discrete Laplace operator $Δ$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\left\|e^{itΔ}\right\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0,$$ 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 $Δ^2$ 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-Schrödinger operators of the form $H=Δ^2+V$ 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↗

arXiv2024

Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials

Let $ H = (-Δ)^m + V $ be a higher-order elliptic operator on $ L^2(\mathbb{R}^n) $, where $ V $ is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with $ H $. In particular, we first establish sharp global Kato smoothing estimates for $ e^{itH} $, based on uniform resolvent estimates of Kato-Yajima type for the absolutely continuous part of $ H $. As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig-Ruiz-Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates.

math.AP↗

arXiv2024

Decay estimates for Beam equations with potentials in dimension three

This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential $$u_{t t}+\big(Δ^2+V\big)u=0, \,\ u(0, x)=f(x),\ u_{t}(0, x)=g(x)$$ in dimension three, where $V$ is a real-valued and decaying potential on $\R^3$. Assume that zero is a regular point of $H:= Δ^2+V $, we first prove the following optimal time decay estimates of the solution operators \begin{equation*} \big\|\cos (t\sqrt{H})P_{ac}(H)\big\|_{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{3}{2}}\ \ \hbox{and} \ \ \Big\|\frac{\sin(t\sqrt{H})}{\sqrt{H}} P_{a c}(H)\Big\|_{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{1}{2}}. \end{equation*} Moreover, if zero is a resonance of $H$, then time decay of the solution operators above also are considered. It is noticed that the first kind resonance does not effect the decay rates for the propagator operators $\cos(t\sqrt{H})$ and $\frac{\sin(t\sqrt{H})}{\sqrt{H}}$, but their decay will be dramatically changed for the second and third resonance types.

math.AP↗

arXiv2024

Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,Δ^2)$ associated with fourth-order Schrödinger operators $H=Δ^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-μ}$ for some $μ>0$. A recent work by Goldberg and Green \cite{GoGr21} has demonstrated that wave operators $W_\pm(H,Δ^2)$ 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 $W_\pm(H,Δ^2)$ in two significant ways. First, we provide counterexamples that illustrate the unboundedness of $W_\pm(H,Δ^2)$ 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 $W_\pm(H,Δ^2)$ and their dual operators $W_\pm(H,Δ^2)^*$ in the case where zero is a regular point and $μ>11$. These estimates depend critically on the singular integral theory of Calderón-Zygmund on a homogeneous space $(X,dω)$ with a doubling measure $dω$.

math.AP↗

arXiv2024

$L^p$-boundedness of wave operators for bi-Schrödinger operators on the line

This paper is devoted to establishing several types of $L^p$-boundedness of wave operators $W_\pm=W_\pm(H, Δ^2)$ associated with the bi-Schrödinger operators $H=Δ^{2}+V(x)$ 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 Hörmander-type $L^p$-boundedness theorem for the spectral multiplier $f(H)$.

math.AP↗

arXiv2023

Decay estimates for fourth-order Schrödinger operators in dimension two

In this paper we study the decay estimates of the fourth order Schrödinger operator $H=Δ^{2}+V(x)$ on $\mathbb{R}^2$ with a bounded decaying potential $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ near the zero threshold in the presence of resonances or eigenvalue, and then use them to establish the $L^1-L^\infty$ decay estimates of $e^{-itH}$generated by the fourth order Schrödinger operator $H$. Our methods used in the decay estimates depend on Littlewood-Paley decomposition and oscillatory integral theory. Moreover, we classify these zero resonances as the distributional solutions of $Hϕ=0$ in suitable weighted spaces. Due to the degeneracy of $Δ^{2}$ at zero threshold and the lower even dimension (i.e. $n=2$), we remark that the asymptotic expansions of resolvent $R_V(λ^4)$ and the classifications of resonances are more involved than Schrödinger operator $-Δ+V$ in dimension two.

math.AP↗

arXiv2022

Stability for the multi-dimensional Borg--Levinson theorem of the biharmonic operator

In this paper, we prove a conditional Hölder stability estimate for the inverse spectral problem of the biharmonic operator. The proof employs the resolvent estimate and a Weyl-type law for the biharmonic operator which were obtained by the authors in \cite{LYZ}. This work extends nontrivially the result in \cite{stefanov} from the second order Schrödinger operator to the fourth order biharmonic operator.

math.AP↗