Convergence of a Class of Schrödinger Equations

Abstract

In this paper, we set up the selection conditions for time series $\{t_k{\}}_{k=1}^{\infty }$ which converge to 0 as $k\to \infty$ such that the solutions of a class of generalized Schrödinger 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 Schrödinger equation when $s<\frac{n}{2(n+1)}$ as $t\to 0$.