Computing Schrödinger propagators on type-2 Turing machines (Q864440)

The computability on various functional spaces can be introduced by means of Type-2 Turing machines and suitable representations. In this framework the paper under review investigates the computability of the solution operators of Schrödinger equations. It is shown that both initial value problems of the linear Schrödinger equation \(u_t = i \Delta u + \phi\) and nonlinear Schrödinger equation \(i u_t = -\Delta u + m u + | u| ^2 u\) have computable solution operators if the initial functions are Sobolev functions. However, if the initial functions are \(L^p\)-functions, then the linear Schrödinger equation has a computable solution operator if and only if \(p= 2\).