Spectral multipliers for Schrödinger operators

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。

对数满足Hölder连续的权重的有效乘子及其在一维薛定谔方程快速边界控制代价中的应用

Effective multipliers for weights whose log are Hölder continuous. Application to the cost of fast boundary controls for the 1D Schr{ö}dinger equation

本文针对对数满足Hölder连续的权重给出了Beurling-Malliavin乘子定理的显式版本，并利用修正Hilbert变换改进了一维Schrödinger方程快速边界控制代价的估计。

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。

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…

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].