We consider the linear and nonlinear cubic Schrödinger equations with periodic boundary conditions and their approximations by splitting methods. We prove that for a dense set of arbitrarily small time steps, there exist numerical solutions leading to strong numerical instabilities, preventing the energy conservation and regularity bounds obtained for the exact solution. We analyze rigorously these instabilities in the semi-discrete and fully discrete cases.
"Resonant time steps and instabilities in the numerical integration of Schrödinger equations." Differential Integral Equations 28 (3/4) 221 - 238, March/April 2015.