Searcharxiv⌕ Search

arXiv subjects

Wenjuan Li

Publications and source records attributed to Wenjuan Li.

33 records · Page 2Linked to original sources
arXiv2023

Analytical results for a spin-orbit coupled atom held in a non-Hermitian double well under synchronous combined modulation

We propose a simple method of synchronous combined modulations to generate the exact analytic solutions for a spin-orbit (SO) coupled ultracold atom held in a non-Hermitian double-well potential. Based on the obtained analytical solutions, we mainly study the parity-time ($\mathcal{PT}$) symmetry of this system and the system stability for both balanced and unbalanced gain-loss between two wells. Under balanced gain and loss, the effect of the proportional constants between synchronous combined modulations and the SO-coupling strength on the $\mathcal{PT}$-symmetry breaking is revealed analytically. Surprisingly, we find when the Zeeman field is present, the stable spin-flipping tunneling between two wells can not occur in the non-Hermitian SO-coupled ultracold atomic system, but the stable spin-conserving tunneling can be performed. Under unbalanced gain and loss, the unique set of parameter conditions that can cause the system to stabilize is found. The results may provide a possibility for the exact control of $\mathcal{PT}$-symmetry breaking and quantum spin dynamics in a non-Hermitian SO-coupled system.

cond-mat.quant-gas↗
arXiv2022

Exact solutions for a spin-orbit coupled bosonic double-well system

Exact solutions for spin-orbit (SO) coupled cold atomic systems are very important and rare in physics. In this paper, we propose a simple method of combined modulations to generate the analytic exact solutions for an SO-coupled boson held in a driven double well. For the cases of synchronous combined modulations and the spin-conserving tunneling, we obtain the general analytical accurate solutions of the system respectively. For the case of spin-flipping tunneling under asynchronous combined modulations, we get the special exact solutions in simple form when the driving parameters satisfy certain conditions. Based on these obtained exact solutions, we reveal some intriguing quantum spin dynamical phenomena, for instance, the arbitrary population transfer (APT) with and/or without spin-flipping, the controlled coherent population conservation (CCPC), and the controlled coherent population inversion (CCPI). The results may have potential applications in the preparation of accurate quantum entangled states and quantum information processing.

quant-ph↗
arXiv2022

Sharp convergence for sequences of Schrödinger means and related generalizations

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, we obtain the almost everywhere convergence results for sequences of Schrödinger means $e^{it_{n}Δ}f$, where $f \in H^{s}(\mathbb{R}^{N}), N\geq 2$. The convergence results are sharp up to the endpoints, and the method can also be applied to get the convergence results for the fractional Schrödinger means and nonelliptic Schrödinger means.

math.CA↗
arXiv2022

Sparse domination and $L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with curvature

In this paper, we study maximal functions along some finite type curves and hypersurfaces. In particular, various impacts of non-isotropic dilations are considered. Firstly, we provide a generic scheme that allows us to deduce the sparse domination bounds for global maximal functions under the assumption that the corresponding localized maximal functions satisfy the $L^{p}$ improving properties. Secondly, for the localized maximal functions with non-isotropic dilations of curves and hypersurfaces whose curvatures vanish to finite order at some points, we establish the $L^{p}\rightarrow L^{q}$ bounds $(q >p)$. As a corollary, we obtain the weighted inequalities for the corresponding global maximal functions, which generalize the known unweighted estimates.

math.CA↗
arXiv2021

Coherent control of spin tunneling in a spin-orbit coupled bosonic triple well

We study the coherent control of spin tunneling for a spin-orbit (SO) coupled boson held in a driven triple well. Under high-frequency approximation, we analytically obtain the quasienergies of the SO-coupled bosonic triple-well system and fine energy band structure is displayed. By adjusting the driving parameters, we reveal that the directed selective spin-flipping or spin-conserving tunneling of a SO-coupled boson occurs along different pathways and in different directions. The analytical results are numerically confirmed and perfect agreements are found. Further, a scheme of quantum spin switch with or without spin-flipping is presented. These results may be useful for quantum information processing and the design of spintronic devices.

physics.atom-ph↗
arXiv2021

On convergence properties for generalized Schrödinger operators along tangential curves

In this paper, we consider convergence properties for generalized Schrödinger operators along tangential curves in $\mathbb{R}^{n} \times \mathbb{R}$ with less smoothness comparing with Lipschitz condition. Firstly, we obtain sharp convergence rate for generalized Schrödinger operators with polynomial growth along tangential curves in $\mathbb{R}^{n} \times \mathbb{R}$, $n \ge 1$. Secondly, it was open until now on pointwise convergence of solutions to the Schrödinger equation along non-$C^1$ curves in $\mathbb{R}^{n} \times \mathbb{R}$, $n\geq 2$, we obtain the corresponding results along some tangential curves when $n=2$ by the broad-narrow argument and polynomial partitioning. Moreover, the corresponding convergence rate will follow. Thirdly, we get the convergence result along a family of restricted tangential curves in $\mathbb{R} \times \mathbb{R}$. As a consequence, we obtain the sharp $L^p$-Schrödinger maximal estimates along tangential curves in $\mathbb{R} \times \mathbb{R}$.

math.CA↗
arXiv2020

Sharp convergence for sequences of nonelliptic Schrödinger means

We consider pointwise convergence of nonelliptic Schrödinger means $e^{it_{n}\square}f(x)$ for $f \in H^{s}(\mathbb{R}^{2})$ and decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, where \[{e^{it_{n}\square }}f\left( x \right): = \int_{{\mathbb{R}^2}} {{e^{i\left( {x \cdot ξ+ t_{n}{{ ξ_{1}ξ_{2} }}} \right)}}\widehat{f}} \left( ξ\right)dξ.\] We prove that when $0<s < \frac{1}{2}$, \[\mathop {\lim }\limits_{n \to \infty} {e^{it_{n}\square }}f\left( x \right) = f(x) \hspace{0.2cm} a.e.\hspace{0.2cm} x\in \mathbb{R}^2\] holds for all $f \in {H^s}\left( {{\mathbb{R}^2}} \right)$ if and only if $\{t_{n}\}_{n=1}^{\infty} \in \ell^{r(s), \infty}(\mathbb{N})$, $r(s)=\frac{s}{1-s}$. Moreover, our result remains valid in general dimensions.

math.CA↗
arXiv2020

Pointwise Convergence for sequences of Schrödinger means in $\mathbb{R}^{2}$

We consider pointwise convergence of Schrödinger means $e^{it_{n}Δ}f(x)$ for $f \in H^{s}(\mathbb{R}^{2})$ and decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero. The main theorem improves the previous results of [Sjölin, JFAA, 2018] and [Sjölin-Strömberg, JMAA, 2020] in $\mathbb{R}^{2}$. This study is based on investigating properties of Schrödinger type maximal functions related to hypersurfaces with vanishing Gaussian curvature.

math.CA↗
arXiv2020

A Note on Non-tangential Convergence for Schrödinger Operators

The goal of this note is to establish non-tangential convergence results for Schrödinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the initial data which implies convergence. As a consequence, we obtain a upper bound for $p$ such that the Schrödinger maximal function is bounded from $H^{s}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ for any $s > \frac{n}{2(n+1)}$.

math.CA↗
arXiv2020

$L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with nonisotropic dilations of hypersurfaces in $\mathbb{R}^3$

The goal of this article is to establish $L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with nonisotropic dilations $δ_t(x)=(t^{a_1}x_1,t^{a_2}x_2,t^{a_3}x_3)$ of hypersurfaces $(x_{1}, x_{2},Φ(x_1,x_2))$ in $\mathbb{R}^3$, where the Gaussian curvatures of the hypersurfaces are allowed to vanish. When $2 α_{2} = α_{3}$, this problem is reduced to study of the $L^{p} \rightarrow L^{q}$ estimates for maximal functions along the curve $γ(x)=(x,x^2(1+ϕ(x)))$ and associated dilations $δ_t(x)=(tx_1,t^2x_2)$. The corresponding maximal function shows features related to the Bourgain circular maximal function, whose $L^{p} \rightarrow L^{q}$ estimate has been considered by [Schlag, JAMS, 1997], [Schlag-Sogge, MRL, 1997] and [Lee, PAMS, 2003]. However, in the study of the maximal function related to the mentioned curve $γ(x)$ and associated dilations, we get the $L^{p} \rightarrow L^{q}$ regularity properties for a family of corresponding Fourier integral operators which fail to satisfy the "cinematic curvature condition" uniformly, which means that classical local smoothing estimates could not be directly applied to our problem. What's more, the $L^{p} \rightarrow L^{q}$ estimates are also new for maximal functions associated with isotropic dilations of hypersurfaces $(x_{1}, x_{2},Φ(x_1,x_2))$ mentioned before.

math.CA↗
arXiv2020

Convergence problems along curves for generalized Schrödinger operators with polynomial growth

In this paper we build the relationship between smoothness of the functions and convergence rate along curves for a class of generalized Schrödinger operators with polynomial growth. We show that the convergence rate depends only on the growth condition of the phase function and regularity of the curve. Our result can be applied to a wide class of operators. In particular, convergence results along curves for a class of generalized Schrödinger operators with non-homogeneous phase functions is built and then the convergence rate is established.

math.CA↗
arXiv2019

A study on a class of generalized Schrödinger operators

In this paper, we consider the pointwise convergence for a class of generalized Schrödinger operators with suitable perturbations, and convergence rate for a class of generalized Schrödinger operators with polynomial growth. We show that the pointwise convergence results remain valid for a class of generalized Schrödinger operators under small perturbations. As applications, we obtain the sharp convergence result for Boussinesq operator and Beam operator in $\mathbb{R}^2$. Moreover, the convergence result for a class of non-elliptic Schrödinger operators with finite-type perturbations is built. Furthermore, we proved that the convergence rate for a class of generalized Schrödinger operators with polynomial growth depends only on the growth condition of their phase functions. This result can be applied to all previously mentioned operators, and more operators.

math.CA↗
arXiv2016

Weighted Caffarelli-Kohn-Nirenberg type inequalities related to Grushin type operators

We consider the Grushin type operator on $\mathbb{R}^{d}_x \times \mathbb{R}^{k}_y$ with the form \begin{equation*} G_μ=\overset{d}{\underset{i=1}{\sum}}\partial_{x_i}^2+\left(\overset{d}{\underset{i=1}{\sum}}x_i^2\right)^{2μ}\overset{k}{\underset{j=1}{\sum}}\partial_{y_j}^2. \end{equation*} and derive weighted Hardy-Sobolev type inequalities and weighted Caffarelli-Kohn-Nirenberg type inequalities related to $G_μ$.

math.AP↗
arXiv2016

Maximal functions associated with nonisotropic dilations of hypersufaces in R^3

The goal of this article is to establish L^p-estimates for maximal functions associated with nonisotropic dilations of hypersurfaces in R^3. Several results have already been obtained by Greenleaf, Iosevich-Sawyer-Seeger, Ikromov-Kempe-Mueller and Zimmermann, but for some situations such as the hypersurface parameterized as the graph of a smooth function $Φ(x_1,x_2)=x_2^d(1+\mathcal{O}(x_2^m))$ near the origin, where $d\geq 2$, $m\geq 1$, and associated dilations $δ_t(x)=(t^ax_1,tx_2,t^dx_3)$ for an arbitrary real number a>0, the question was open until recently. In fact, such problems do arise already in lower dimensions. For instance, we consider the curve $γ(x)=(x,x^2(1+ϕ(x)))$ and associated dilations $δ_t(x)=(tx_1,t^2x_2)$. If $ϕ\equiv 0$, then the corresponding maximal function is the maximal function along parabolas in the plane, which is very well understood due to the work by Nagel-Riviere-Wainger and others. If $ϕ\neq 0$ and $ϕ(x)=\mathcal{O}(x^m)$, $m\geq 1$, the problem was open until recently, however, the corresponding maximal function shows features related to the Bourgain circular maximal function, which required deep ideas and local smoothing estimates established by Mockenhaupt-Seeger-Sogge for Fourier integral operators satisfying the so-called "cinematic curvature" condition. However, we observe that in the study of M related to the mentioned curve $γ(x)$ and associated dilations, we will consider a family of corresponding Fourier integral operators which fail to satisfy the "cinematic curvature condition" uniformly, which means that classical local smoothing estimates could not be directly applied to our problem. In this article, we develop new ideas in order to overcome the above difficulty and finally establish sharp L^p-estimates for the maximal function related to the curve $γ(x)$ ......

math.CA↗