Bilinear estimates associated to the Schrödinger equation with a nonelliptic principal part

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 …

Schrödinger方程解的相位恢复

Phase retrieval for solutions of the Schr{ö}dinger equations

本文提出Schrödinger方程相位恢复的若干问题，给出部分解答及启发式思考。

On unique continuation for solutions of the Schr{ö}dinger equation on trees

We prove that if a solution of the time-dependent Schr{ö}dinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schr{ö}dinger operator, we use the spectral theory of the Laplacian and complex analysis and obtain a characterization of the initial conditions that lead to a sharp decay at any time. We then use the recent spectral decomposition of the Schr{ö}dinger operator with compactly supported potential due to Colin de Verdi{è}rre and Turc to extend our results in the presence of such potentials. Finally, we u…

Blow-up of solutions to the nonlinear Schr$\bf{\ddot{O}}$dinger equations on standard N-sphere and hyperbolic N-space

In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr$\ddot{o}$dinger equations on some Riemannian manifolds like the standard 2-sphere $S^2$ and the hyperbolic 2-space $H^{2}(-1)$. Using the similar idea, we establish such blow-up results on higher dimensional standard sphere and hyperbolic $n$-space. Extensions to $n$-dimensional Riemannian warped product manifolds with $n\geq 2$ are also given.

关于某些周期图上离散 Schrödinger 算子的三球不等式

On the Three Balls Inequality for Discrete Schr{ö}dinger Operators on Certain Periodic Graphs

本文利用 Carleman 估计研究周期图上离散磁性 Schrödinger 算子的定量唯一延拓，并将三球不等式推广至包含六角晶格的周期图族。

An uncertainty principle for solutions of the Schr{ö}dinger equation on H-type groups

In this paper we consider uncertainty principles for solutions of certain PDEs on H-type groups. We first prove that, contrary to the euclidean setting, the heat kernel on H-type groups is not characterized as the only solution of the heat equation that has sharp decay at 2 different times. We then prove the analogue of Hardy's Uncertainty Principle for solutions of the Schr{ö}dinger equation with potential on H-type groups. This extends the free case considered by Ben Saïd, Dogga and Thangavelu [BTD] and by Ludwig and M{ü}ller [LM].