Regular propagators of bilinear quantum systems

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

We study optimal bilinear control problems for stochastic nonlinear Schrödinger equations in both the mass subcritical and critical case. For general initial data of the minimal L2 regularity, we prove the existence and first order Lagrange condition of an open loop control. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type U2 and V2, adapted to evolution operators related to linear Schrödinger equations with lower order perturbations. In particular, we obtain a new temporal regularity of rescaled (backward) stochastic solutions, which is the key…

A sharp regularity estimate for the Schrödinger propagator on the sphere

Let $Δ_{\mathbb S^n}$ denote the Laplace-Beltrami operator on the $n$-dimensional unit sphere $\mathbb S^n$. In this paper we show that $$ \| e^{it Δ_{\mathbb S^n}}f \|_{L^4([0, 2π) \times \mathbb S^n)} \leq C \| f\|_{W^{α, 4} (\mathbb S^n)} $$ holds provided that $n\geq 2$, $α> {(n-2)/4}.$ The range of $α$ is sharp up to the endpoint. As a consequence, we obtain space-time estimates for the Schrödinger propagator $e^{it Δ_{\mathbb S^n}}$ on the $L^p$ spaces for $2\leq p\leq \infty.$ We also prove that for zonal functions on ${\mathbb S}^n$, the Schrödinger maximal operator $\sup_{0\leq t<2π} …

A Seeger-Sogge-Stein theorem for bilinear Fourier integral operators

We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in $S^m_{1,0} (n,2)$ and non-degenerate phase functions, from $L^p \times L^q \to L^r$ under the assumptions that $m\leq -(n-1)(|\frac{1}{p}-\frac{1}{2}|+|\frac{1}{q}-\frac{1}{2}|)$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$. This is a bilinear version of the classical theorem of Seeger-Sogge-Stein concerning the $L^p$ boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of non-Banach target spaces.

带导数非线性项与准周期初值的色散方程的低正则适定性

Low-regularity well-posedness for dispersive equations with derivative nonlinearity and quasi-periodic initial data

该研究证明了带导数非线性项与准周期初值的Korteweg-de Vries方程的低正则适定性，通过频率依赖时间局部化等方法推导双线性Strichartz估计，得到的解可保持初值的Sobolev正则性，且论证可推广至其他色散关系及高阶非线性项。

Sharp bilinear estimates and well-posedness for the 1-D Schrödinger-Debye system

We establish local and global well-posedness for the initial value problem associated to the one-dimensional Schrodinger-Debye (SD) system for data in the Sobolev spaces with low regularity. To obtain local results we prove two new sharp bilinear estimates for the coupling terms of this system in the continuous and periodic cases. Concerning global results, in the continuous case, the system is shown to be globally well-posed in $H^s\times H^s, -3/14< s< 0$. For initial data in Sobolev spaces with high regularity ($H^s\times H^s, s>5/2$), Bidégaray \cite{Bidegaray} proved that there are one-pa…

Efficient finite dimensional approximations for the bilinear Schrodinger equation with bounded variation controls

This the text of a proceeding accepted for the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014). We present some results of an ongoing research on the controllability problem of an abstract bilinear Schrodinger equation. We are interested by approximation of this equation by finite dimensional systems. Assuming that the uncontrolled term $A$ has a pure discrete spectrum and the control potential $B$ is in some sense regular with respect to $A$ we show that such an approximation is possible. More precisely the solutions are approximated by their projection…

带权和不带权的双线性伪微分算子在一般Hörmander类中的正则性

Weighted and unweighted regularity of bilinear pseudo-differential operators with symbols in general Hörmander classes

本文研究了在未探索的 regime 中双线性伪微分算子的有界性，通过精确的最大函数建立了算子的点wise 估计，并扩展了参数范围，获得了双线性伪微分算子在多线性 A_{p,q,r} 权下的加权norm不等式。

通过双线性电磁场实现 $\mathbb{R}^N$ 上非线性 Schrödinger 方程的小时间能控性

Small-time controllability for the nonlinear Schrödinger equation on $\mathbb{R}^N$ via bilinear electromagnetic fields

本文研究 $\mathbb{R}^N$ 上受磁、电外场作用的 NLS 的小时间能控性，借助无限维传播子的非交换性证明双线性方程能控，并给出零时间能量调控结果。

Bilinear embedding for divergence-form operators with complex coefficients on irregular domains

Let $Ω\subseteq \mathbb{R}^{d}$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $Ω$ with $L^{\infty}$ coefficients. Consider the divergence-form operator ${\mathscr L}^{A}=-{\rm div}(A\nabla)$ with mixed boundary conditions on $Ω$. We extend the bilinear inequality that we proved in [16] in the special case when $Ω=\mathbb{R}^{d}$. As a consequence, we obtain that the solution to the parabolic problem $u^{\prime}(t)+{\mathscr L}^{A}u(t)=f(t)$, $u(0)=0$, has maximal regularity in $L^{p}(Ω)$, for all $p>1$ such that $A$ satisfies the $p$-ellipticity c…

有限正则性下因果传播子的Sobolev波前集

The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity

本文研究有限正则性整体双曲时空下Klein-Gordon算子因果传播子的Sobolev波前集，给出不同正则性条件下波前集的包含与等式关系，并验证传播子整体正则性与光滑情形一致。

Constructive solution of a bilinear optimal control problem for a Schrödinger equation

Often considered in numerical simulations related to the control of quantum systems, the so-called monotonic schemes have not been so far much studied from the functional analysis point of view. Yet, these procedures provide an efficient constructive method for solving a certain class of optimal control problems. This paper aims both at extending the results already available about these algorithms in the finite dimensional case (i.e., the time-discretized case) and at completing those of the continuous case. This paper starts with some results about the regularity of a functional related to a…

Sharp local well-posedness for the "good" Boussinesq equation

In the present article, we prove the sharp local well-posedness and ill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; the initial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ and ill-posed in $H^s(\mathbb{T})$ for $s<-1/2$. Well-posedness result is obtained from reduction of the problem into a quadratic nonlinear Schrödinger equation and the contraction argument in suitably modified $X^{s,b}$ spaces. The proof of the crucial bilinear estimates in these spaces, especially in the lowest regularity, rely on some bilinear estimates for one dimensional perio…

On Gevrey Regularity of the Supercritical SQG equation in Critical Besov Spaces

In this paper we show that the solution of the supercrti- cal surface quasi-geostrophic (SQG) equation, starting from initial data in homogeneous critical Besov spaces belong to a subanalytic Gevrey class. In particular, we improve upon the result of Dong and Li in [26], where they showed that the solutions of Chen-Miao-Zhang (cf. [11]) are classical solutions. We extend the approach of Biswas (cf. [7]) to critical, L^p -based Besov spaces, and adapt the point of view of Lemarie- Rieusset (cf. [36]), who treated the operator arising from applying the analytic Gevrey operator to a product of an…

抽象分数阶柯西问题：传播子的存在性与非齐次解表示

Abstract Fractional Cauchy Problem: Existence of Propagators and Inhomogeneous Solution Representation

本文针对由无界线性算子和 Caputo 分数阶导数定义的非齐次柯西问题，提出仅依赖齐次问题传播子的新温和解表示，统一了次抛物、抛物和次双曲情形，并在显著更弱的正则性假设下证明强收敛，适用于任意阶数 α∈(0,2) 的数值计算。