Computing Schrödinger propagators on Type-2 Turing machines

We study Turing computability of the solution operators of the initial-value problems for the linear Schrödinger equation ut=iΔu+φ and the nonlinear Schrödinger equation of the form iut=-Δu+mu+|u|2u. We prove that the solution operators are computable if the initial data are Sobolev functions but noncomputable in the linear case if the initial data are Lp-functions and p≠2. The computations are performed on Type-2 Turing machines.