Pointwise convergence of solutions to Schrödinger type equations

In this paper, we study the pointwise convergence of the function eitΔα2f(x) whenever ff belongs to some Sobolev space. When α=2α=2, eitΔf(x)eitΔf(x) is the solution to the standard Schrödinger equation and this problem has been widely studied. Here we focus on the case 0<α≠10<α≠1. The same convergence problem for the nonelliptic Schrödinger equation is also discussed. The proof here follows the idea of Carleson’s first paper on this subject. But we do reprove some known theorems by this straightforward method and get some new results as well.