Related papers: An improved necessary condition for the Schr\"odin…
One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition $f$ of the Schr\"odinger equation given by \begin{equation*} \begin{cases}…
In this paper, we study the almost everywhere convergence of the cubic nonlinear Schr\"odinger flow to the initial data on $\mathbb S^2$, \begin{equation*} iu_t + \Delta_g u = |u|^2u, \quad (t,x)\in\R\times \S^2. \end{equation*} Inspired by…
We study the large-time behavior of the solutions to the Schr\"odinger equation associated with a quickly decaying potential in dimension three. We establish the resolvent expansions at threshold zero and near positive resonances. The…
We obtain global well-posedness, scattering, and global $L^{\frac{2(n+2)}{n-2}}_{t,x}$ spacetime bounds for energy-space solutions to the energy-critical nonlinear Schr\"odinger equation in $\R_t\times \R^n_x$, $n\geq 5$.
We investigate the pointwise convergence of the solution to the fractional Schr\"odinger equation in $\mathbb R^2$. By establishing $H^s(\mathbb R^2)-L^3(\mathbb R^2)$ estimates for the associated maximal operator provided that $s>1/3$, we…
This note presents a new proof of the well-known Strichartz estimates for the Schr\"odinger equation in $2+1$ dimensions, building on ideas from our recent work \cite{MO}.
We consider the Schr\"odinger equation on a half space in any dimension with a class of nonhomogeneous boundary conditions including Dirichlet, Neuman and the so-called transparent boundary conditions. Building upon recent local in time…
We prove global well-posedness for the cubic, defocusing, nonlinear Schr{\"o}dinger equation on $\mathbf{R}^{2}$ with data $u_{0} \in H^{s}(\mathbf{R}^{2})$, $s > 1/4$. We accomplish this by improving the almost Morawetz estimates in [9].
Maximal estimates for Schr\"odinger means and convergence almost everywhere of sequences of Schr\"odinger means are studied.
In \cite{poiret}, we explain how we can construct global solutions for the cubic Schr\"odinger equation in three dimensional with initial data in $ L^2(\mathds{R}^3) $. The main ingredient of this proof is the existence of the bilinear…
The initial value problem for the $L^{2}$ critical semilinear Schr\"odinger equation in $\R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^{s}({\Bbb R^{n}})$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$, and…
In this paper, we consider the maximal estimates for the solution to an initial value problem of the linear Schroedinger equation with a singular potential. We show a result about the pointwise convergence of solutions to this special…
A new type of solution for the full 3+1 dimensional space-time Schroedinger equation is presented here. We consider elegant presentation of the exact solution in a spherical coordinate system, along with the assuming of separation of the…
Almost everywhere convergence on the solution of Schr\"odinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this problem was essentially solved by building the sharp $L^p$-estimate of…
We consider the pointwise convergence problem for the solution of Schr\"odinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson's problem as the most simple case and was…
We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schr\"odinger and wave equations under the assumption of $H^1$ solutions. For the Schr\"odinger equation we analyze…
A long standing conjecture for the linear Schroedinger equation states that 1/4 of derivative in $L^2$, in the sense of Sobolev spaces, suffices in any dimension for the solution to that equation to converge almost everywhere to the initial…
We reconsider the Schr\"oder-Siegel problem of conjugating an analytic map in $\mathbb{C}$ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension $n>1$. Assuming a condition which is equivalent to…
Let $\mathcal S^2$ be the Stepanov space and let $ \lambda_n\uparrow\infty$. Let $(a_n)_{n\ge 1}$ be satisfying Wiener's condition $A:= \sum_{n\ge 1} \big(\sum_{k\, :\, n\le \lambda_k \le n+1}|a_k|\big)^2 <\infty$. We prove that $\big\|…
We present a generalization of the often-used Crank-Nicolson (CN) method of obtaining numerical solutions of the time-dependent Schr\"odinger equation. The generalization yields numerical solutions accurate to order $(\Delta x)^{2r-1}$ in…