On the link between Schrodinger and Heisenberg pictures of quantum mechanics

The authors take into account boundary conditions in solutions of the Heisenberg equations of motion. They express the kernel of the evolution operator in the x-representation through the retarded propagator which solves the Schrodinger equation and satisfies the boundary conditions in question. As an illustration, they solve the Heisenberg equation for a free particle bound to the half-line (0, infinity ) with various boundary conditions at the point x=0.