Related papers: Pointwise Convergence for sequences of Schr\"{o}di…

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, we obtain the almost everywhere convergence results for sequences of Schr\"{o}dinger means $e^{it_{n}\Delta}f$, where $f \in H^{s}(\mathbb{R}^{N}), N\geq 2$. The…

We consider pointwise convergence of nonelliptic Schr\"{o}dinger 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…

We study pointwise convergence of the fractional Schr\"odinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-\Delta)^{\alpha/2}} f| $ can be deduced from those…

For functions in the Sobolev space $H^s$ and decreasing sequences $t_n\to 0$ we examine convergence almost everywhere of the generalized Schr\"odinger means on the real line, given by \[S^af(x,t_n)=\exp( it_n (-\partial_{xx})^{a/2})f(x);\]…

We study pointwise convergence of the solutions to Schr\"odinger equations with initial datum $f\in H^s(\mathbb R^n)$. The conjecture is that the solution $e^{it\Delta}f$ converges to $f$ almost everywhere for all $f\in H^s(\mathbb R^n)$ if…

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…

In this paper, we set up the selection conditions for time series $\{t_k\}_{k=1}^\infty$ which converge to 0 as $k\rightarrow\infty$ such that the solutions of a class of generalized Schr\"odinger equations almost everywhere pointwise…

We obtain partial improvement toward the pointwise convergence problem of Schr\"odinger solutions, in the general setting of fractal measure. In particular, we show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ almost…

We study the Hausdorff dimension of the sets on which the pointwise convergence of the solutions to the fractional Schr\"odinger equation $e^{it(-\Delta)^\frac m2}f$ fails when $m\in(0,1)$ in one spatial dimension. The pointwise convergence…

We study the pointwise convergence of solutions to the free Schr\"{o}dinger equation with initial data in the Bessel potential spaces $L_s^p(\mathbb{R}^n)$. We establish new sufficient regularity indices for pointwise convergence across the…

The goal of this note is to establish non-tangential convergence results for Schr\"{o}dinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the…

We show that $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.

We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is…

In this paper we address the question of the pointwise almost everywhere limit of nonlinear Schr\"odinger flows to the initial data, in both the continuous and the periodic settings. Then we show how, in some cases, certain smoothing…

We study convergence almost everywhere of sequences of Schr\"odinger means. We also replace sequences by uncountable sets.

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…

In this paper, we establish the almost everywhere convergence of solutions to the Schr\"odinger operator with complex time $ P_{\gamma}f(x,t) $ in higher dimensions, under the assumption that the initial data $f$ belongs to the Sobolev…

In \cite{Lee:2006:schrod-converg}, when the spatial variable $x$ is localized, Lee observed that the Schr\"odinger maximal operator $e^{it\Delta}f(x)$ enjoys certain localization property in $t$ for frequency localized functions. In this…

In this paper, we consider the convergence problem of Schr\"odinger equation. Firstly, we show the almost everywhere pointwise convergence of Schr\"odinger equation in Fourier-Lebesgue spaces…

We investigate the Sobolev regularity required for almost everywhere convergence to the initial datum of solutions to the linear Schr\"odinger equation along certain tangential curves. In the regime $\alpha<\tfrac12$, we analyze maximal…