A Survey on the Study of Generalized Schrödinger Operators along Curves

In this survey, we review the historical development for the Carleson problem about the a.e. pointwise convergence in five aspects: the a.e. convergence for generalized Schrödinger operators along vertical lines; a.e. convergence for Schrödinger operators along arbitrary single curves; a.e. convergence for Schrödinger operators along a family of restricted curves; upper bounds of p for the Lp-Schrödinger maximal estimates; and a.e. convergence rate for generalized Schrödinger operators along curves. Finally, we list some open problems which need to be addressed.