分数阶噪声驱动的随机非线性Schrodinger方程的适定性、大偏差和支撑

On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations

Under certain regularity conditions, we establish quasi-invariance of Gaussian measures on periodic functions under the flow of cubic fractional nonlinear Schrödinger equations on the one-dimensional torus.

Location of maximizers of eigenfunctions of fractional Schrödinger's equation

Eigenfunctions of the fractional Schrödinger operators in a domain $\mathcal{D}$ are considered, and a relation between the supremum of the potential and the distance of a maximizer of the eigenfunction from $\partial\mathcal{D}$ is established. This, in particular, extends a recent result of Rachh and Steinerberger to the fractional Schrödinger operators. We also propose a fractional version of the Barta's inequality and also generalize a celebrated Lieb's theorem for fractional Schrödinger operators. As applications of above results we obtain a Faber-Krahn inequality for non-local Schrödinge…

带线性漂移的随机薛定谔方程的中偏差

Moderate Deviations for a Stochastic Schrödinger Equation with Linear Drift

本文通过弱收敛方法研究了带线性漂移项和$Q$-维纳过程驱动的随机薛定谔型方程，实现了中偏差原理并证明了中心极限定理以分析其渐近行为。

分数阶Schrodinger算子的迹渐近性

Trace Asymptotics for Fractional Schrodinger Operators

本文证明了Banuelos和Sa Barreto关于$\R^d$上Schrodinger算子迹的渐近展开的结果的类似结果，当布朗运动的生成元拉普拉斯$Δ$被非局部积分算子$Δ^{α/2}$、$0<α<2$（它是$α$阶对称稳定过程的生成元）取代时。这些结果也扩展了Banuelos和Yildirim最近的结果，其中计算了$Δ^{α/2}$的前两个系数。得到了相对论稳定过程和混合稳定过程中薛定谔算子的一些推广。

时间分数阶薛定谔方程

Time fractional Schrodinger equation

将薛定谔方程的一阶时间导数转化为Caputo分数阶导数，即时间分数阶薛定谔方程。由此产生的哈密顿量被发现是非埃尔米特和非本地的时间。由此产生的波函数在时间反演下不是不变的。时间分数薛定谔方程求解的一个自由粒子和一个势阱。发现概率和由此产生的能级随时间增加到取决于时间导数的阶数的极限值。Mittag-Leffler函数的新身份也被发现并在附录中提出。