Convergence of a Class of Schr\"{o}dinger Equations

Abstract

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 converge to their initial data in $H^s(\mathbb{R}^n)$ for $s>0$. As it is known that the pointwise convergence can not be true for Schr\"odinger equation when $s<\frac{n}{2(n+1)}$ as $t\rightarrow0$.