It is well known that the heat diffusion equation
$$ u_t - u_{xx} = 0 ,\quad u(0, x ) = f(x) , $$
has the smoothing property.
The question is, how about the imaginary equivalent of it, namely the Schrodinger equation?
$$ i u_t = u_{xx} ,\quad u(0, x ) = f(x) . $$
If the initial state $f$ is not derivative at some point, is this property retained by the solution $u$?
