The paper is a survey of recent results on a.e. convergence of the solutionu(x, t) to the Schrodinger equation\(\Delta u = i\frac{{\partial u}}{{\partial t}}\) andu(x, 0)=f(x) wheref belongs to the Schwartz class Open image in new window . A.e. convergence is established via estimates of the kind $$\left( {\int\limits_B {(\mathop {Sup}\limits_{0< t< 1} |u(x,t)|)^2 dx} } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \leqslant C_B \left\| f \right\|_{H_8 } $$ whereB is a ball in Open image in new window andH, denotes the classical Sobolev space.