Approximate propagator for the Schrödinger equation (Q1571203)

The author considers the Cauchy problem for the Schrödinger equation: \[ i\hbar\partial_t\Psi=-(1/2)\hbar^2\nabla^2\Psi+\vee\Psi,\quad \Psi(\cdot,+0)=0. \tag{1} \] Let \(U(t)\) be the operator associating the initial data of problem (1), with the solution to (1) at the point \(t\). An approximate propagator for the Schrödinger equation is constructed by means of Gaussian wave packets, i.e. an operator \(J(t,\delta{t})\) is derived such that, on some class of functions, it satisfies the estimate \[ \|[U(t)-J(t,\delta{t})]\|\leq c\delta t^{1+\alpha},\quad \alpha>0. \]